%------------------------------------------------------------------------------
% 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    : VYDRA_MDL/0009_Window/prob_01725_078563__16549154_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2218 ( 749 unt; 931 typ;   0 def)
%            Number of atoms       : 2952 (1809 equ;   0 cnn)
%            Maximal formula atoms :   14 (   2 avg)
%            Number of connectives : 13323 ( 431   ~;  11   |; 178   &;11592   @)
%                                         (   0 <=>;1111  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (   6 avg)
%            Number of types       :  138 ( 137 usr)
%            Number of type conns  : 3503 (3503   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  797 ( 794 usr;  44 con; 0-5 aty)
%            Number of variables   : 4323 ( 660   ^;3591   !;  72   ?;4323   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:53:24.567
%------------------------------------------------------------------------------
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    comp_P8916405840549426341_c_nat: ( product_prod_b_c > b ) > ( produc6689720525226317995_c_nat > product_prod_b_c ) > produc6689720525226317995_c_nat > b ).

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    comp_P680640439754360793_nat_c: ( product_prod_c_nat > nat ) > ( product_prod_nat_c > product_prod_c_nat ) > product_prod_nat_c > nat ).

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    comp_P1418391719297794398_c_nat: ( product_prod_c_nat > produc4862256710654508797_c_nat ) > ( produc4862256710654508797_c_nat > product_prod_c_nat ) > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat ).

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    comp_P1391869959443764803_nat_c: ( product_prod_c_nat > c ) > ( product_prod_nat_c > product_prod_c_nat ) > product_prod_nat_c > c ).

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thf(sy_c_Window_Ommap__map_001tf__b_001t__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_001tf__c,type,
    mmap_m3523663605641305108_nat_c: ( b > produc4862256710654508797_c_nat > c ) > list_P7417839048565863355_c_nat > list_P903359562653991662od_b_c ).

thf(sy_c_Window_Ommap__map_001tf__b_001tf__c_001t__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J,type,
    mmap_m8776140404295941026_c_nat: ( b > c > option7520157102916957007_c_nat ) > list_P903359562653991662od_b_c > list_P8155366108056616461_c_nat ).

thf(sy_c_Window_Ommap__map_001tf__b_001tf__c_001t__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J,type,
    mmap_m3921707120754670672_c_nat: ( b > c > produc4862256710654508797_c_nat ) > list_P903359562653991662od_b_c > list_P7417839048565863355_c_nat ).

thf(sy_c_Window_Ommap__map_001tf__b_001tf__c_001tf__c,type,
    mmap_map_b_c_c: ( b > c > c ) > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c ).

thf(sy_c_Window_Ommap__update_001tf__b_001t__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J,type,
    mmap_u5371215486469568154_c_nat: b > option7520157102916957007_c_nat > list_P8155366108056616461_c_nat > list_P8155366108056616461_c_nat ).

thf(sy_c_Window_Ommap__update_001tf__b_001t__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J,type,
    mmap_u2662500248198004552_c_nat: b > produc4862256710654508797_c_nat > list_P7417839048565863355_c_nat > list_P7417839048565863355_c_nat ).

thf(sy_c_Window_Ommap__update_001tf__b_001tf__c,type,
    mmap_update_b_c: b > c > list_P903359562653991662od_b_c > list_P903359562653991662od_b_c ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_J,type,
    member8863226656820029924_c_nat: option2970150418924381261_c_nat > set_op6736743094888940163_c_nat > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J,type,
    member3208607818295432806_c_nat: option7520157102916957007_c_nat > set_op6398572775360143365_c_nat > $o ).

thf(sy_c_member_001t__Option__Ooption_Itf__b_J,type,
    member_option_b: option_b > set_option_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__c_J,type,
    member8962352060716553276_nat_c: product_prod_nat_c > set_Pr4335409928928841947_nat_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J,type,
    member6296142003466748712_c_nat: produc3510132715374707071_c_nat > set_Pr4914646408830571231_c_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_Mtf__b_J,type,
    member1023106406135730172_nat_b: produc7545861912598698981_nat_b > set_Pr1066409585067842203_nat_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_J,type,
    member16691529365684868_c_nat: produc4895531831895483099_c_nat > set_Pr6545832687900118075_c_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J,type,
    member871170437899409800_c_nat: produc5740701590023533791_c_nat > set_Pr8671814610322996543_c_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__b_J_Mt__Set__Oset_Itf__b_J_J,type,
    member318967379524898064_set_b: produc3262564261791608551_set_b > set_Pr7275202699945397959_set_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__List__Olist_It__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_J_J,type,
    member9033503127983833042_c_nat: produc6358558248090350139_c_nat > set_Pr55366283650883953_c_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_Mtf__b_J_J,type,
    member8160214478689597918_nat_b: produc3246764621125188789_nat_b > set_Pr3745380843212982421_nat_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J,type,
    member7562873241046315796_c_nat: produc4862256710654508797_c_nat > set_Pr8806432033423503795_c_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_J,type,
    member1522478204950355394_c_nat: produc6689720525226317995_c_nat > set_Pr93524137074774369_c_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    member7862447936710763792od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__c_J,type,
    member7862447941013992593od_b_c: product_prod_b_c > set_Product_prod_b_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J,type,
    member8195077246299207702_c_nat: product_prod_c_nat > set_Pr6903500605879609269_c_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mtf__c_J,type,
    member5074992359041316560od_c_c: product_prod_c_c > set_Product_prod_c_c > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
    member_set_b: set_b > set_set_b > $o ).

thf(sy_c_member_001tf__b,type,
    member_b: b > set_b > $o ).

thf(sy_c_member_001tf__c,type,
    member_c: c > set_c > $o ).

thf(sy_v_e_H____,type,
    e: list_P903359562653991662od_b_c ).

thf(sy_v_e____,type,
    e2: list_P903359562653991662od_b_c ).

thf(sy_v_init____,type,
    init: b ).

thf(sy_v_q__cur____,type,
    q_cur: b ).

thf(sy_v_s_H_H____,type,
    s: list_P7417839048565863355_c_nat ).

thf(sy_v_s_H____,type,
    s2: list_P7417839048565863355_c_nat ).

thf(sy_v_s____,type,
    s3: list_P7417839048565863355_c_nat ).

thf(sy_v_s__cur____,type,
    s_cur: list_P7417839048565863355_c_nat ).

thf(sy_v_t____,type,
    t: c ).

thf(sy_v_tstp__cur____,type,
    tstp_cur: option7520157102916957007_c_nat ).

% Relevant facts (1269)
thf(fact_0__092_060open_062s_H_H_A_092_060equiv_062_Ammap__update_Ainit_A_Icase_Ammap__lookup_As__cur_Aq__cur_Aof_ANone_A_092_060Rightarrow_062_A_Iq__cur_M_Atstp__cur_J_A_124_ASome_A_Iq_H_M_Atstp_H_J_A_092_060Rightarrow_062_Acase_Atstp_H_Aof_ANone_A_092_060Rightarrow_062_A_Iq_H_M_Atstp__cur_J_A_124_ASome_A_Its_M_Atp_J_A_092_060Rightarrow_062_A_Iq_H_M_Atstp_H_J_J_As_H_092_060close_062,axiom,
    ( s
    = ( mmap_u2662500248198004552_c_nat @ init
      @ ( case_o1602653449097480839_c_nat @ ( produc5716802255202478839_c_nat @ q_cur @ tstp_cur )
        @ ( produc3722540595886809633_c_nat
          @ ^ [Q: b,Tstp: option7520157102916957007_c_nat] :
              ( case_o6498799718091186953_c_nat @ ( produc5716802255202478839_c_nat @ Q @ tstp_cur )
              @ ( produc3293230613241633187_c_nat
                @ ^ [Ts: c,Tp: nat] : ( produc5716802255202478839_c_nat @ Q @ Tstp ) )
              @ Tstp ) )
        @ ( mmap_l5026999719965937591_c_nat @ s_cur @ q_cur ) )
      @ s2 ) ) ).

% \<open>s'' \<equiv> mmap_update init (case mmap_lookup s_cur q_cur of None \<Rightarrow> (q_cur, tstp_cur) | Some (q', tstp') \<Rightarrow> case tstp' of None \<Rightarrow> (q', tstp_cur) | Some (ts, tp) \<Rightarrow> (q', tstp')) s'\<close>
thf(fact_1_None,axiom,
    ( ( mmap_l5026999719965937591_c_nat @ s_cur @ q_cur )
    = none_P8487441334512977628_c_nat ) ).

% None
thf(fact_2_s_H_H__def,axiom,
    ( s
    = ( mmap_u2662500248198004552_c_nat @ init
      @ ( case_o1602653449097480839_c_nat @ ( produc5716802255202478839_c_nat @ q_cur @ tstp_cur )
        @ ( produc3722540595886809633_c_nat
          @ ^ [Q: b,Tstp: option7520157102916957007_c_nat] :
              ( case_o6498799718091186953_c_nat @ ( produc5716802255202478839_c_nat @ Q @ tstp_cur )
              @ ( produc3293230613241633187_c_nat
                @ ^ [Ts: c,Tp: nat] : ( produc5716802255202478839_c_nat @ Q @ Tstp ) )
              @ Tstp ) )
        @ ( mmap_l5026999719965937591_c_nat @ s_cur @ q_cur ) )
      @ s2 ) ) ).

% s''_def
thf(fact_3_case__prod__conv,axiom,
    ! [F: b > option7520157102916957007_c_nat > b,A: b,B: option7520157102916957007_c_nat] :
      ( ( produc9009988077629141723_nat_b @ F @ ( produc5716802255202478839_c_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_4_case__prod__conv,axiom,
    ! [F: b > c > product_prod_c_b,A: b,B: c] :
      ( ( produc6717796504295817721od_c_b @ F @ ( product_Pair_b_c @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5_case__prod__conv,axiom,
    ! [F: b > c > $o,A: b,B: c] :
      ( ( produc1873627588981547243_b_c_o @ F @ ( product_Pair_b_c @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_6_case__prod__conv,axiom,
    ! [F: b > c > b,A: b,B: c] :
      ( ( produc8837194378803474130_b_c_b @ F @ ( product_Pair_b_c @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_7_case__prod__conv,axiom,
    ! [F: b > b > b,A: b,B: b] :
      ( ( produc2401277923921374355_b_b_b @ F @ ( product_Pair_b_b @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_8_case__prod__conv,axiom,
    ! [F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,A: b,B: option7520157102916957007_c_nat] :
      ( ( produc3722540595886809633_c_nat @ F @ ( produc5716802255202478839_c_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_9_case__prod__conv,axiom,
    ! [F: c > nat > produc4862256710654508797_c_nat,A: c,B: nat] :
      ( ( produc3293230613241633187_c_nat @ F @ ( product_Pair_c_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_10_case__prod__Pair__iden,axiom,
    ! [P: produc6689720525226317995_c_nat] :
      ( ( produc5367230572582240381_c_nat @ produc8868828497325608613_c_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_11_case__prod__Pair__iden,axiom,
    ! [P: produc3246764621125188789_nat_b] :
      ( ( produc6284714251336508411_nat_b @ produc8656096559574149925_nat_b @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_12_case__prod__Pair__iden,axiom,
    ! [P: produc6358558248090350139_c_nat] :
      ( ( produc7964627196075380253_c_nat @ produc385356081875622325_c_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_13_case__prod__Pair__iden,axiom,
    ! [P: product_prod_b_c] :
      ( ( produc281880053716946747od_b_c @ product_Pair_b_c @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_14_case__prod__Pair__iden,axiom,
    ! [P: product_prod_b_b] :
      ( ( produc921320349745534075od_b_b @ product_Pair_b_b @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_15_case__prod__Pair__iden,axiom,
    ! [P: product_prod_c_nat] :
      ( ( produc1049061115736377381_c_nat @ product_Pair_c_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_16_case__prod__Pair__iden,axiom,
    ! [P: produc4862256710654508797_c_nat] :
      ( ( produc3722540595886809633_c_nat @ produc5716802255202478839_c_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_17_prod_Oinject,axiom,
    ! [X1: b,X2: produc4862256710654508797_c_nat,Y1: b,Y2: produc4862256710654508797_c_nat] :
      ( ( ( produc8868828497325608613_c_nat @ X1 @ X2 )
        = ( produc8868828497325608613_c_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_18_prod_Oinject,axiom,
    ! [X1: b,X2: mappin7688773591249790157_nat_b,Y1: b,Y2: mappin7688773591249790157_nat_b] :
      ( ( ( produc8656096559574149925_nat_b @ X1 @ X2 )
        = ( produc8656096559574149925_nat_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_19_prod_Oinject,axiom,
    ! [X1: b,X2: list_P8155366108056616461_c_nat,Y1: b,Y2: list_P8155366108056616461_c_nat] :
      ( ( ( produc385356081875622325_c_nat @ X1 @ X2 )
        = ( produc385356081875622325_c_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_20_prod_Oinject,axiom,
    ! [X1: b,X2: c,Y1: b,Y2: c] :
      ( ( ( product_Pair_b_c @ X1 @ X2 )
        = ( product_Pair_b_c @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_21_prod_Oinject,axiom,
    ! [X1: b,X2: b,Y1: b,Y2: b] :
      ( ( ( product_Pair_b_b @ X1 @ X2 )
        = ( product_Pair_b_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_22_prod_Oinject,axiom,
    ! [X1: c,X2: nat,Y1: c,Y2: nat] :
      ( ( ( product_Pair_c_nat @ X1 @ X2 )
        = ( product_Pair_c_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_23_prod_Oinject,axiom,
    ! [X1: b,X2: option7520157102916957007_c_nat,Y1: b,Y2: option7520157102916957007_c_nat] :
      ( ( ( produc5716802255202478839_c_nat @ X1 @ X2 )
        = ( produc5716802255202478839_c_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_24_old_Oprod_Oinject,axiom,
    ! [A: b,B: produc4862256710654508797_c_nat,A2: b,B2: produc4862256710654508797_c_nat] :
      ( ( ( produc8868828497325608613_c_nat @ A @ B )
        = ( produc8868828497325608613_c_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_25_old_Oprod_Oinject,axiom,
    ! [A: b,B: mappin7688773591249790157_nat_b,A2: b,B2: mappin7688773591249790157_nat_b] :
      ( ( ( produc8656096559574149925_nat_b @ A @ B )
        = ( produc8656096559574149925_nat_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_26_old_Oprod_Oinject,axiom,
    ! [A: b,B: list_P8155366108056616461_c_nat,A2: b,B2: list_P8155366108056616461_c_nat] :
      ( ( ( produc385356081875622325_c_nat @ A @ B )
        = ( produc385356081875622325_c_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_27_old_Oprod_Oinject,axiom,
    ! [A: b,B: c,A2: b,B2: c] :
      ( ( ( product_Pair_b_c @ A @ B )
        = ( product_Pair_b_c @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_28_old_Oprod_Oinject,axiom,
    ! [A: b,B: b,A2: b,B2: b] :
      ( ( ( product_Pair_b_b @ A @ B )
        = ( product_Pair_b_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_29_old_Oprod_Oinject,axiom,
    ! [A: c,B: nat,A2: c,B2: nat] :
      ( ( ( product_Pair_c_nat @ A @ B )
        = ( product_Pair_c_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_30_old_Oprod_Oinject,axiom,
    ! [A: b,B: option7520157102916957007_c_nat,A2: b,B2: option7520157102916957007_c_nat] :
      ( ( ( produc5716802255202478839_c_nat @ A @ B )
        = ( produc5716802255202478839_c_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_31_case__prodE2,axiom,
    ! [Q2: b > $o,P2: b > option7520157102916957007_c_nat > b,Z: produc4862256710654508797_c_nat] :
      ( ( Q2 @ ( produc9009988077629141723_nat_b @ P2 @ Z ) )
     => ~ ! [X: b,Y: option7520157102916957007_c_nat] :
            ( ( Z
              = ( produc5716802255202478839_c_nat @ X @ Y ) )
           => ~ ( Q2 @ ( P2 @ X @ Y ) ) ) ) ).

% case_prodE2
thf(fact_32_case__prodE2,axiom,
    ! [Q2: product_prod_c_b > $o,P2: b > c > product_prod_c_b,Z: product_prod_b_c] :
      ( ( Q2 @ ( produc6717796504295817721od_c_b @ P2 @ Z ) )
     => ~ ! [X: b,Y: c] :
            ( ( Z
              = ( product_Pair_b_c @ X @ Y ) )
           => ~ ( Q2 @ ( P2 @ X @ Y ) ) ) ) ).

% case_prodE2
thf(fact_33_case__prodE2,axiom,
    ! [Q2: $o > $o,P2: b > c > $o,Z: product_prod_b_c] :
      ( ( Q2 @ ( produc1873627588981547243_b_c_o @ P2 @ Z ) )
     => ~ ! [X: b,Y: c] :
            ( ( Z
              = ( product_Pair_b_c @ X @ Y ) )
           => ~ ( Q2 @ ( P2 @ X @ Y ) ) ) ) ).

% case_prodE2
thf(fact_34_case__prodE2,axiom,
    ! [Q2: b > $o,P2: b > c > b,Z: product_prod_b_c] :
      ( ( Q2 @ ( produc8837194378803474130_b_c_b @ P2 @ Z ) )
     => ~ ! [X: b,Y: c] :
            ( ( Z
              = ( product_Pair_b_c @ X @ Y ) )
           => ~ ( Q2 @ ( P2 @ X @ Y ) ) ) ) ).

% case_prodE2
thf(fact_35_case__prodE2,axiom,
    ! [Q2: b > $o,P2: b > b > b,Z: product_prod_b_b] :
      ( ( Q2 @ ( produc2401277923921374355_b_b_b @ P2 @ Z ) )
     => ~ ! [X: b,Y: b] :
            ( ( Z
              = ( product_Pair_b_b @ X @ Y ) )
           => ~ ( Q2 @ ( P2 @ X @ Y ) ) ) ) ).

% case_prodE2
thf(fact_36_case__prodE2,axiom,
    ! [Q2: produc4862256710654508797_c_nat > $o,P2: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,Z: produc4862256710654508797_c_nat] :
      ( ( Q2 @ ( produc3722540595886809633_c_nat @ P2 @ Z ) )
     => ~ ! [X: b,Y: option7520157102916957007_c_nat] :
            ( ( Z
              = ( produc5716802255202478839_c_nat @ X @ Y ) )
           => ~ ( Q2 @ ( P2 @ X @ Y ) ) ) ) ).

% case_prodE2
thf(fact_37_case__prodE2,axiom,
    ! [Q2: produc4862256710654508797_c_nat > $o,P2: c > nat > produc4862256710654508797_c_nat,Z: product_prod_c_nat] :
      ( ( Q2 @ ( produc3293230613241633187_c_nat @ P2 @ Z ) )
     => ~ ! [X: c,Y: nat] :
            ( ( Z
              = ( product_Pair_c_nat @ X @ Y ) )
           => ~ ( Q2 @ ( P2 @ X @ Y ) ) ) ) ).

% case_prodE2
thf(fact_38_case__prod__eta,axiom,
    ! [F: produc4862256710654508797_c_nat > b] :
      ( ( produc9009988077629141723_nat_b
        @ ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( F @ ( produc5716802255202478839_c_nat @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_39_case__prod__eta,axiom,
    ! [F: product_prod_b_c > product_prod_c_b] :
      ( ( produc6717796504295817721od_c_b
        @ ^ [X3: b,Y3: c] : ( F @ ( product_Pair_b_c @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_40_case__prod__eta,axiom,
    ! [F: product_prod_b_c > $o] :
      ( ( produc1873627588981547243_b_c_o
        @ ^ [X3: b,Y3: c] : ( F @ ( product_Pair_b_c @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_41_case__prod__eta,axiom,
    ! [F: product_prod_b_c > b] :
      ( ( produc8837194378803474130_b_c_b
        @ ^ [X3: b,Y3: c] : ( F @ ( product_Pair_b_c @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_42_case__prod__eta,axiom,
    ! [F: product_prod_b_b > b] :
      ( ( produc2401277923921374355_b_b_b
        @ ^ [X3: b,Y3: b] : ( F @ ( product_Pair_b_b @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_43_case__prod__eta,axiom,
    ! [F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( produc3722540595886809633_c_nat
        @ ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( F @ ( produc5716802255202478839_c_nat @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_44_case__prod__eta,axiom,
    ! [F: product_prod_c_nat > produc4862256710654508797_c_nat] :
      ( ( produc3293230613241633187_c_nat
        @ ^ [X3: c,Y3: nat] : ( F @ ( product_Pair_c_nat @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_45_cond__case__prod__eta,axiom,
    ! [F: b > option7520157102916957007_c_nat > b,G: produc4862256710654508797_c_nat > b] :
      ( ! [X: b,Y: option7520157102916957007_c_nat] :
          ( ( F @ X @ Y )
          = ( G @ ( produc5716802255202478839_c_nat @ X @ Y ) ) )
     => ( ( produc9009988077629141723_nat_b @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_46_cond__case__prod__eta,axiom,
    ! [F: b > c > product_prod_c_b,G: product_prod_b_c > product_prod_c_b] :
      ( ! [X: b,Y: c] :
          ( ( F @ X @ Y )
          = ( G @ ( product_Pair_b_c @ X @ Y ) ) )
     => ( ( produc6717796504295817721od_c_b @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_47_cond__case__prod__eta,axiom,
    ! [F: b > c > $o,G: product_prod_b_c > $o] :
      ( ! [X: b,Y: c] :
          ( ( F @ X @ Y )
          = ( G @ ( product_Pair_b_c @ X @ Y ) ) )
     => ( ( produc1873627588981547243_b_c_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_48_cond__case__prod__eta,axiom,
    ! [F: b > c > b,G: product_prod_b_c > b] :
      ( ! [X: b,Y: c] :
          ( ( F @ X @ Y )
          = ( G @ ( product_Pair_b_c @ X @ Y ) ) )
     => ( ( produc8837194378803474130_b_c_b @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_49_cond__case__prod__eta,axiom,
    ! [F: b > b > b,G: product_prod_b_b > b] :
      ( ! [X: b,Y: b] :
          ( ( F @ X @ Y )
          = ( G @ ( product_Pair_b_b @ X @ Y ) ) )
     => ( ( produc2401277923921374355_b_b_b @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_50_cond__case__prod__eta,axiom,
    ! [F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,G: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ! [X: b,Y: option7520157102916957007_c_nat] :
          ( ( F @ X @ Y )
          = ( G @ ( produc5716802255202478839_c_nat @ X @ Y ) ) )
     => ( ( produc3722540595886809633_c_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_51_cond__case__prod__eta,axiom,
    ! [F: c > nat > produc4862256710654508797_c_nat,G: product_prod_c_nat > produc4862256710654508797_c_nat] :
      ( ! [X: c,Y: nat] :
          ( ( F @ X @ Y )
          = ( G @ ( product_Pair_c_nat @ X @ Y ) ) )
     => ( ( produc3293230613241633187_c_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_52_old_Oprod_Ocase,axiom,
    ! [F: b > option7520157102916957007_c_nat > b,X1: b,X2: option7520157102916957007_c_nat] :
      ( ( produc9009988077629141723_nat_b @ F @ ( produc5716802255202478839_c_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_53_old_Oprod_Ocase,axiom,
    ! [F: b > c > product_prod_c_b,X1: b,X2: c] :
      ( ( produc6717796504295817721od_c_b @ F @ ( product_Pair_b_c @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_54_old_Oprod_Ocase,axiom,
    ! [F: b > c > $o,X1: b,X2: c] :
      ( ( produc1873627588981547243_b_c_o @ F @ ( product_Pair_b_c @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_55_old_Oprod_Ocase,axiom,
    ! [F: b > c > b,X1: b,X2: c] :
      ( ( produc8837194378803474130_b_c_b @ F @ ( product_Pair_b_c @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_56_old_Oprod_Ocase,axiom,
    ! [F: b > b > b,X1: b,X2: b] :
      ( ( produc2401277923921374355_b_b_b @ F @ ( product_Pair_b_b @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_57_old_Oprod_Ocase,axiom,
    ! [F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,X1: b,X2: option7520157102916957007_c_nat] :
      ( ( produc3722540595886809633_c_nat @ F @ ( produc5716802255202478839_c_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_58_old_Oprod_Ocase,axiom,
    ! [F: c > nat > produc4862256710654508797_c_nat,X1: c,X2: nat] :
      ( ( produc3293230613241633187_c_nat @ F @ ( product_Pair_c_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_59_split__cong,axiom,
    ! [Q3: produc4862256710654508797_c_nat,F: b > option7520157102916957007_c_nat > b,G: b > option7520157102916957007_c_nat > b,P: produc4862256710654508797_c_nat] :
      ( ! [X: b,Y: option7520157102916957007_c_nat] :
          ( ( ( produc5716802255202478839_c_nat @ X @ Y )
            = Q3 )
         => ( ( F @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q3 )
       => ( ( produc9009988077629141723_nat_b @ F @ P )
          = ( produc9009988077629141723_nat_b @ G @ Q3 ) ) ) ) ).

% split_cong
thf(fact_60_split__cong,axiom,
    ! [Q3: product_prod_b_c,F: b > c > product_prod_c_b,G: b > c > product_prod_c_b,P: product_prod_b_c] :
      ( ! [X: b,Y: c] :
          ( ( ( product_Pair_b_c @ X @ Y )
            = Q3 )
         => ( ( F @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q3 )
       => ( ( produc6717796504295817721od_c_b @ F @ P )
          = ( produc6717796504295817721od_c_b @ G @ Q3 ) ) ) ) ).

% split_cong
thf(fact_61_split__cong,axiom,
    ! [Q3: product_prod_b_c,F: b > c > $o,G: b > c > $o,P: product_prod_b_c] :
      ( ! [X: b,Y: c] :
          ( ( ( product_Pair_b_c @ X @ Y )
            = Q3 )
         => ( ( F @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q3 )
       => ( ( produc1873627588981547243_b_c_o @ F @ P )
          = ( produc1873627588981547243_b_c_o @ G @ Q3 ) ) ) ) ).

% split_cong
thf(fact_62_split__cong,axiom,
    ! [Q3: product_prod_b_c,F: b > c > b,G: b > c > b,P: product_prod_b_c] :
      ( ! [X: b,Y: c] :
          ( ( ( product_Pair_b_c @ X @ Y )
            = Q3 )
         => ( ( F @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q3 )
       => ( ( produc8837194378803474130_b_c_b @ F @ P )
          = ( produc8837194378803474130_b_c_b @ G @ Q3 ) ) ) ) ).

% split_cong
thf(fact_63_split__cong,axiom,
    ! [Q3: product_prod_b_b,F: b > b > b,G: b > b > b,P: product_prod_b_b] :
      ( ! [X: b,Y: b] :
          ( ( ( product_Pair_b_b @ X @ Y )
            = Q3 )
         => ( ( F @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q3 )
       => ( ( produc2401277923921374355_b_b_b @ F @ P )
          = ( produc2401277923921374355_b_b_b @ G @ Q3 ) ) ) ) ).

% split_cong
thf(fact_64_split__cong,axiom,
    ! [Q3: produc4862256710654508797_c_nat,F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,G: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,P: produc4862256710654508797_c_nat] :
      ( ! [X: b,Y: option7520157102916957007_c_nat] :
          ( ( ( produc5716802255202478839_c_nat @ X @ Y )
            = Q3 )
         => ( ( F @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q3 )
       => ( ( produc3722540595886809633_c_nat @ F @ P )
          = ( produc3722540595886809633_c_nat @ G @ Q3 ) ) ) ) ).

% split_cong
thf(fact_65_split__cong,axiom,
    ! [Q3: product_prod_c_nat,F: c > nat > produc4862256710654508797_c_nat,G: c > nat > produc4862256710654508797_c_nat,P: product_prod_c_nat] :
      ( ! [X: c,Y: nat] :
          ( ( ( product_Pair_c_nat @ X @ Y )
            = Q3 )
         => ( ( F @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q3 )
       => ( ( produc3293230613241633187_c_nat @ F @ P )
          = ( produc3293230613241633187_c_nat @ G @ Q3 ) ) ) ) ).

% split_cong
thf(fact_66_option_Ocase__distrib,axiom,
    ! [H: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,F1: produc4862256710654508797_c_nat,F2: product_prod_c_nat > produc4862256710654508797_c_nat,Option: option7520157102916957007_c_nat] :
      ( ( H @ ( case_o6498799718091186953_c_nat @ F1 @ F2 @ Option ) )
      = ( case_o6498799718091186953_c_nat @ ( H @ F1 )
        @ ^ [X3: product_prod_c_nat] : ( H @ ( F2 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_67_option_Ocase__distrib,axiom,
    ! [H: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,F1: produc4862256710654508797_c_nat,F2: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Option: option2970150418924381261_c_nat] :
      ( ( H @ ( case_o1602653449097480839_c_nat @ F1 @ F2 @ Option ) )
      = ( case_o1602653449097480839_c_nat @ ( H @ F1 )
        @ ^ [X3: produc4862256710654508797_c_nat] : ( H @ ( F2 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_68_option_Ocase__distrib,axiom,
    ! [H: $o > $o,F1: $o,F2: b > $o,Option: option_b] :
      ( ( H @ ( case_option_o_b @ F1 @ F2 @ Option ) )
      = ( case_option_o_b @ ( H @ F1 )
        @ ^ [X3: b] : ( H @ ( F2 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_69_option_Ocase__distrib,axiom,
    ! [H: nat > nat,F1: nat,F2: nat > nat,Option: option_nat] :
      ( ( H @ ( case_option_nat_nat @ F1 @ F2 @ Option ) )
      = ( case_option_nat_nat @ ( H @ F1 )
        @ ^ [X3: nat] : ( H @ ( F2 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_70_option_Ocase__distrib,axiom,
    ! [H: c > c,F1: c,F2: c > c,Option: option_c] :
      ( ( H @ ( case_option_c_c @ F1 @ F2 @ Option ) )
      = ( case_option_c_c @ ( H @ F1 )
        @ ^ [X3: c] : ( H @ ( F2 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_71_option_Ocase__distrib,axiom,
    ! [H: $o > $o,F1: $o,F2: product_prod_c_nat > $o,Option: option7520157102916957007_c_nat] :
      ( ( H @ ( case_o3625425862587997644_c_nat @ F1 @ F2 @ Option ) )
      = ( case_o3625425862587997644_c_nat @ ( H @ F1 )
        @ ^ [X3: product_prod_c_nat] : ( H @ ( F2 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_72_option_Ocase__distrib,axiom,
    ! [H: $o > $o,F1: $o,F2: produc4862256710654508797_c_nat > $o,Option: option2970150418924381261_c_nat] :
      ( ( H @ ( case_o5420598178156702154_c_nat @ F1 @ F2 @ Option ) )
      = ( case_o5420598178156702154_c_nat @ ( H @ F1 )
        @ ^ [X3: produc4862256710654508797_c_nat] : ( H @ ( F2 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_73_option_Ocase__distrib,axiom,
    ! [H: produc4862256710654508797_c_nat > $o,F1: produc4862256710654508797_c_nat,F2: product_prod_c_nat > produc4862256710654508797_c_nat,Option: option7520157102916957007_c_nat] :
      ( ( H @ ( case_o6498799718091186953_c_nat @ F1 @ F2 @ Option ) )
      = ( case_o3625425862587997644_c_nat @ ( H @ F1 )
        @ ^ [X3: product_prod_c_nat] : ( H @ ( F2 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_74_option_Ocase__distrib,axiom,
    ! [H: $o > produc4862256710654508797_c_nat,F1: $o,F2: product_prod_c_nat > $o,Option: option7520157102916957007_c_nat] :
      ( ( H @ ( case_o3625425862587997644_c_nat @ F1 @ F2 @ Option ) )
      = ( case_o6498799718091186953_c_nat @ ( H @ F1 )
        @ ^ [X3: product_prod_c_nat] : ( H @ ( F2 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_75_option_Ocase__distrib,axiom,
    ! [H: produc4862256710654508797_c_nat > $o,F1: produc4862256710654508797_c_nat,F2: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Option: option2970150418924381261_c_nat] :
      ( ( H @ ( case_o1602653449097480839_c_nat @ F1 @ F2 @ Option ) )
      = ( case_o5420598178156702154_c_nat @ ( H @ F1 )
        @ ^ [X3: produc4862256710654508797_c_nat] : ( H @ ( F2 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_76_disjE__realizer2,axiom,
    ! [P2: $o,Q2: nat > $o,X4: option_nat,R: nat > $o,F: nat,G: nat > nat] :
      ( ( case_option_o_nat @ P2 @ Q2 @ X4 )
     => ( ( P2
         => ( R @ F ) )
       => ( ! [Q4: nat] :
              ( ( Q2 @ Q4 )
             => ( R @ ( G @ Q4 ) ) )
         => ( R @ ( case_option_nat_nat @ F @ G @ X4 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_77_disjE__realizer2,axiom,
    ! [P2: $o,Q2: c > $o,X4: option_c,R: c > $o,F: c,G: c > c] :
      ( ( case_option_o_c @ P2 @ Q2 @ X4 )
     => ( ( P2
         => ( R @ F ) )
       => ( ! [Q4: c] :
              ( ( Q2 @ Q4 )
             => ( R @ ( G @ Q4 ) ) )
         => ( R @ ( case_option_c_c @ F @ G @ X4 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_78_disjE__realizer2,axiom,
    ! [P2: $o,Q2: product_prod_c_nat > $o,X4: option7520157102916957007_c_nat,R: $o > $o,F: $o,G: product_prod_c_nat > $o] :
      ( ( case_o3625425862587997644_c_nat @ P2 @ Q2 @ X4 )
     => ( ( P2
         => ( R @ F ) )
       => ( ! [Q4: product_prod_c_nat] :
              ( ( Q2 @ Q4 )
             => ( R @ ( G @ Q4 ) ) )
         => ( R @ ( case_o3625425862587997644_c_nat @ F @ G @ X4 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_79_disjE__realizer2,axiom,
    ! [P2: $o,Q2: produc4862256710654508797_c_nat > $o,X4: option2970150418924381261_c_nat,R: $o > $o,F: $o,G: produc4862256710654508797_c_nat > $o] :
      ( ( case_o5420598178156702154_c_nat @ P2 @ Q2 @ X4 )
     => ( ( P2
         => ( R @ F ) )
       => ( ! [Q4: produc4862256710654508797_c_nat] :
              ( ( Q2 @ Q4 )
             => ( R @ ( G @ Q4 ) ) )
         => ( R @ ( case_o5420598178156702154_c_nat @ F @ G @ X4 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_80_disjE__realizer2,axiom,
    ! [P2: $o,Q2: b > $o,X4: option_b,R: $o > $o,F: $o,G: b > $o] :
      ( ( case_option_o_b @ P2 @ Q2 @ X4 )
     => ( ( P2
         => ( R @ F ) )
       => ( ! [Q4: b] :
              ( ( Q2 @ Q4 )
             => ( R @ ( G @ Q4 ) ) )
         => ( R @ ( case_option_o_b @ F @ G @ X4 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_81_disjE__realizer2,axiom,
    ! [P2: $o,Q2: produc4862256710654508797_c_nat > $o,X4: option2970150418924381261_c_nat,R: produc4862256710654508797_c_nat > $o,F: produc4862256710654508797_c_nat,G: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( case_o5420598178156702154_c_nat @ P2 @ Q2 @ X4 )
     => ( ( P2
         => ( R @ F ) )
       => ( ! [Q4: produc4862256710654508797_c_nat] :
              ( ( Q2 @ Q4 )
             => ( R @ ( G @ Q4 ) ) )
         => ( R @ ( case_o1602653449097480839_c_nat @ F @ G @ X4 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_82_disjE__realizer2,axiom,
    ! [P2: $o,Q2: product_prod_c_nat > $o,X4: option7520157102916957007_c_nat,R: produc4862256710654508797_c_nat > $o,F: produc4862256710654508797_c_nat,G: product_prod_c_nat > produc4862256710654508797_c_nat] :
      ( ( case_o3625425862587997644_c_nat @ P2 @ Q2 @ X4 )
     => ( ( P2
         => ( R @ F ) )
       => ( ! [Q4: product_prod_c_nat] :
              ( ( Q2 @ Q4 )
             => ( R @ ( G @ Q4 ) ) )
         => ( R @ ( case_o6498799718091186953_c_nat @ F @ G @ X4 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_83_prod_Ocase__distrib,axiom,
    ! [H: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,F: c > nat > produc4862256710654508797_c_nat,Prod: product_prod_c_nat] :
      ( ( H @ ( produc3293230613241633187_c_nat @ F @ Prod ) )
      = ( produc3293230613241633187_c_nat
        @ ^ [X12: c,X22: nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_84_prod_Ocase__distrib,axiom,
    ! [H: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,Prod: produc4862256710654508797_c_nat] :
      ( ( H @ ( produc3722540595886809633_c_nat @ F @ Prod ) )
      = ( produc3722540595886809633_c_nat
        @ ^ [X12: b,X22: option7520157102916957007_c_nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_85_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: b > c > $o,Prod: product_prod_b_c] :
      ( ( H @ ( produc1873627588981547243_b_c_o @ F @ Prod ) )
      = ( produc1873627588981547243_b_c_o
        @ ^ [X12: b,X22: c] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_86_prod_Ocase__distrib,axiom,
    ! [H: $o > b,F: b > c > $o,Prod: product_prod_b_c] :
      ( ( H @ ( produc1873627588981547243_b_c_o @ F @ Prod ) )
      = ( produc8837194378803474130_b_c_b
        @ ^ [X12: b,X22: c] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_87_prod_Ocase__distrib,axiom,
    ! [H: b > $o,F: b > c > b,Prod: product_prod_b_c] :
      ( ( H @ ( produc8837194378803474130_b_c_b @ F @ Prod ) )
      = ( produc1873627588981547243_b_c_o
        @ ^ [X12: b,X22: c] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_88_prod_Ocase__distrib,axiom,
    ! [H: b > b,F: b > c > b,Prod: product_prod_b_c] :
      ( ( H @ ( produc8837194378803474130_b_c_b @ F @ Prod ) )
      = ( produc8837194378803474130_b_c_b
        @ ^ [X12: b,X22: c] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_89_prod_Ocase__distrib,axiom,
    ! [H: b > b,F: b > b > b,Prod: product_prod_b_b] :
      ( ( H @ ( produc2401277923921374355_b_b_b @ F @ Prod ) )
      = ( produc2401277923921374355_b_b_b
        @ ^ [X12: b,X22: b] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_90_prod_Ocase__distrib,axiom,
    ! [H: product_prod_c_b > $o,F: b > c > product_prod_c_b,Prod: product_prod_b_c] :
      ( ( H @ ( produc6717796504295817721od_c_b @ F @ Prod ) )
      = ( produc1873627588981547243_b_c_o
        @ ^ [X12: b,X22: c] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_91_prod_Ocase__distrib,axiom,
    ! [H: product_prod_c_b > b,F: b > c > product_prod_c_b,Prod: product_prod_b_c] :
      ( ( H @ ( produc6717796504295817721od_c_b @ F @ Prod ) )
      = ( produc8837194378803474130_b_c_b
        @ ^ [X12: b,X22: c] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_92_prod_Ocase__distrib,axiom,
    ! [H: $o > product_prod_c_b,F: b > c > $o,Prod: product_prod_b_c] :
      ( ( H @ ( produc1873627588981547243_b_c_o @ F @ Prod ) )
      = ( produc6717796504295817721od_c_b
        @ ^ [X12: b,X22: c] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_93_case__prodI2,axiom,
    ! [P: produc6689720525226317995_c_nat,C: b > produc4862256710654508797_c_nat > $o] :
      ( ! [A3: b,B3: produc4862256710654508797_c_nat] :
          ( ( P
            = ( produc8868828497325608613_c_nat @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc6484547981188837958_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_94_case__prodI2,axiom,
    ! [P: produc3246764621125188789_nat_b,C: b > mappin7688773591249790157_nat_b > $o] :
      ( ! [A3: b,B3: mappin7688773591249790157_nat_b] :
          ( ( P
            = ( produc8656096559574149925_nat_b @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc5419437825036861918at_b_o @ C @ P ) ) ).

% case_prodI2
thf(fact_95_case__prodI2,axiom,
    ! [P: produc6358558248090350139_c_nat,C: b > list_P8155366108056616461_c_nat > $o] :
      ( ! [A3: b,B3: list_P8155366108056616461_c_nat] :
          ( ( P
            = ( produc385356081875622325_c_nat @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc9156697109305850166_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_96_case__prodI2,axiom,
    ! [P: product_prod_b_b,C: b > b > $o] :
      ( ! [A3: b,B3: b] :
          ( ( P
            = ( product_Pair_b_b @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc4772849988445941548_b_b_o @ C @ P ) ) ).

% case_prodI2
thf(fact_97_case__prodI2,axiom,
    ! [P: product_prod_b_c,C: b > c > $o] :
      ( ! [A3: b,B3: c] :
          ( ( P
            = ( product_Pair_b_c @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc1873627588981547243_b_c_o @ C @ P ) ) ).

% case_prodI2
thf(fact_98_case__prodI2,axiom,
    ! [P: produc4862256710654508797_c_nat,C: b > option7520157102916957007_c_nat > $o] :
      ( ! [A3: b,B3: option7520157102916957007_c_nat] :
          ( ( P
            = ( produc5716802255202478839_c_nat @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc2544960137906461044_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_99_case__prodI2,axiom,
    ! [P: product_prod_c_nat,C: c > nat > $o] :
      ( ! [A3: c,B3: nat] :
          ( ( P
            = ( product_Pair_c_nat @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc7133942929724870258_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_100_case__prodI,axiom,
    ! [F: b > produc4862256710654508797_c_nat > $o,A: b,B: produc4862256710654508797_c_nat] :
      ( ( F @ A @ B )
     => ( produc6484547981188837958_nat_o @ F @ ( produc8868828497325608613_c_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_101_case__prodI,axiom,
    ! [F: b > mappin7688773591249790157_nat_b > $o,A: b,B: mappin7688773591249790157_nat_b] :
      ( ( F @ A @ B )
     => ( produc5419437825036861918at_b_o @ F @ ( produc8656096559574149925_nat_b @ A @ B ) ) ) ).

% case_prodI
thf(fact_102_case__prodI,axiom,
    ! [F: b > list_P8155366108056616461_c_nat > $o,A: b,B: list_P8155366108056616461_c_nat] :
      ( ( F @ A @ B )
     => ( produc9156697109305850166_nat_o @ F @ ( produc385356081875622325_c_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_103_case__prodI,axiom,
    ! [F: b > b > $o,A: b,B: b] :
      ( ( F @ A @ B )
     => ( produc4772849988445941548_b_b_o @ F @ ( product_Pair_b_b @ A @ B ) ) ) ).

% case_prodI
thf(fact_104_case__prodI,axiom,
    ! [F: b > c > $o,A: b,B: c] :
      ( ( F @ A @ B )
     => ( produc1873627588981547243_b_c_o @ F @ ( product_Pair_b_c @ A @ B ) ) ) ).

% case_prodI
thf(fact_105_case__prodI,axiom,
    ! [F: b > option7520157102916957007_c_nat > $o,A: b,B: option7520157102916957007_c_nat] :
      ( ( F @ A @ B )
     => ( produc2544960137906461044_nat_o @ F @ ( produc5716802255202478839_c_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_106_case__prodI,axiom,
    ! [F: c > nat > $o,A: c,B: nat] :
      ( ( F @ A @ B )
     => ( produc7133942929724870258_nat_o @ F @ ( product_Pair_c_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_107_mem__case__prodI2,axiom,
    ! [P: product_prod_c_nat,Z: b,C: c > nat > set_b] :
      ( ! [A3: c,B3: nat] :
          ( ( P
            = ( product_Pair_c_nat @ A3 @ B3 ) )
         => ( member_b @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_b @ Z @ ( produc454361093202844089_set_b @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_108_mem__case__prodI2,axiom,
    ! [P: produc4862256710654508797_c_nat,Z: b,C: b > option7520157102916957007_c_nat > set_b] :
      ( ! [A3: b,B3: option7520157102916957007_c_nat] :
          ( ( P
            = ( produc5716802255202478839_c_nat @ A3 @ B3 ) )
         => ( member_b @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_b @ Z @ ( produc4086673573166660539_set_b @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_109_mem__case__prodI2,axiom,
    ! [P: product_prod_c_nat,Z: nat,C: c > nat > set_nat] :
      ( ! [A3: c,B3: nat] :
          ( ( P
            = ( product_Pair_c_nat @ A3 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc4234106380533341996et_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_110_mem__case__prodI2,axiom,
    ! [P: product_prod_c_nat,Z: c,C: c > nat > set_c] :
      ( ! [A3: c,B3: nat] :
          ( ( P
            = ( product_Pair_c_nat @ A3 @ B3 ) )
         => ( member_c @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_c @ Z @ ( produc454361097506072890_set_c @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_111_mem__case__prodI2,axiom,
    ! [P: product_prod_b_c,Z: b,C: b > c > set_b] :
      ( ! [A3: b,B3: c] :
          ( ( P
            = ( product_Pair_b_c @ A3 @ B3 ) )
         => ( member_b @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_b @ Z @ ( produc3958908711191305138_set_b @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_112_mem__case__prodI2,axiom,
    ! [P: product_prod_b_c,Z: nat,C: b > c > set_nat] :
      ( ! [A3: b,B3: c] :
          ( ( P
            = ( product_Pair_b_c @ A3 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc8716184571927494899et_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_113_mem__case__prodI2,axiom,
    ! [P: product_prod_b_c,Z: c,C: b > c > set_c] :
      ( ! [A3: b,B3: c] :
          ( ( P
            = ( product_Pair_b_c @ A3 @ B3 ) )
         => ( member_c @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_c @ Z @ ( produc3958908715494533939_set_c @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_114_mem__case__prodI2,axiom,
    ! [P: product_prod_b_b,Z: b,C: b > b > set_b] :
      ( ! [A3: b,B3: b] :
          ( ( P
            = ( product_Pair_b_b @ A3 @ B3 ) )
         => ( member_b @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_b @ Z @ ( produc2614443674381817203_set_b @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_115_mem__case__prodI2,axiom,
    ! [P: product_prod_b_b,Z: nat,C: b > b > set_nat] :
      ( ! [A3: b,B3: b] :
          ( ( P
            = ( product_Pair_b_b @ A3 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc5893810021816043762et_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_116_mem__case__prodI2,axiom,
    ! [P: product_prod_b_b,Z: c,C: b > b > set_c] :
      ( ! [A3: b,B3: b] :
          ( ( P
            = ( product_Pair_b_b @ A3 @ B3 ) )
         => ( member_c @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_c @ Z @ ( produc2614443678685046004_set_c @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_117_mem__case__prodI,axiom,
    ! [Z: b,C: c > nat > set_b,A: c,B: nat] :
      ( ( member_b @ Z @ ( C @ A @ B ) )
     => ( member_b @ Z @ ( produc454361093202844089_set_b @ C @ ( product_Pair_c_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_118_mem__case__prodI,axiom,
    ! [Z: b,C: b > option7520157102916957007_c_nat > set_b,A: b,B: option7520157102916957007_c_nat] :
      ( ( member_b @ Z @ ( C @ A @ B ) )
     => ( member_b @ Z @ ( produc4086673573166660539_set_b @ C @ ( produc5716802255202478839_c_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_119_mem__case__prodI,axiom,
    ! [Z: nat,C: c > nat > set_nat,A: c,B: nat] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc4234106380533341996et_nat @ C @ ( product_Pair_c_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_120_mem__case__prodI,axiom,
    ! [Z: c,C: c > nat > set_c,A: c,B: nat] :
      ( ( member_c @ Z @ ( C @ A @ B ) )
     => ( member_c @ Z @ ( produc454361097506072890_set_c @ C @ ( product_Pair_c_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_121_mem__case__prodI,axiom,
    ! [Z: b,C: b > c > set_b,A: b,B: c] :
      ( ( member_b @ Z @ ( C @ A @ B ) )
     => ( member_b @ Z @ ( produc3958908711191305138_set_b @ C @ ( product_Pair_b_c @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_122_mem__case__prodI,axiom,
    ! [Z: nat,C: b > c > set_nat,A: b,B: c] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc8716184571927494899et_nat @ C @ ( product_Pair_b_c @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_123_mem__case__prodI,axiom,
    ! [Z: c,C: b > c > set_c,A: b,B: c] :
      ( ( member_c @ Z @ ( C @ A @ B ) )
     => ( member_c @ Z @ ( produc3958908715494533939_set_c @ C @ ( product_Pair_b_c @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_124_mem__case__prodI,axiom,
    ! [Z: b,C: b > b > set_b,A: b,B: b] :
      ( ( member_b @ Z @ ( C @ A @ B ) )
     => ( member_b @ Z @ ( produc2614443674381817203_set_b @ C @ ( product_Pair_b_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_125_mem__case__prodI,axiom,
    ! [Z: nat,C: b > b > set_nat,A: b,B: b] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc5893810021816043762et_nat @ C @ ( product_Pair_b_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_126_mem__case__prodI,axiom,
    ! [Z: c,C: b > b > set_c,A: b,B: b] :
      ( ( member_c @ Z @ ( C @ A @ B ) )
     => ( member_c @ Z @ ( produc2614443678685046004_set_c @ C @ ( product_Pair_b_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_127_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option6424319895471668383_c_nat] :
      ( ( Option != none_o1329286346417566254_c_nat )
      = ( case_o658134151661089564_c_nat @ $false
        @ ^ [Uu: option7520157102916957007_c_nat] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_128_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_c] :
      ( ( Option != none_c )
      = ( case_option_o_c @ $false
        @ ^ [Uu: c] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_129_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
      = ( case_option_o_nat @ $false
        @ ^ [Uu: nat] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_130_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option7520157102916957007_c_nat] :
      ( ( Option != none_P6573378090780909534_c_nat )
      = ( case_o3625425862587997644_c_nat @ $false
        @ ^ [Uu: product_prod_c_nat] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_131_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_b] :
      ( ( Option != none_b )
      = ( case_option_o_b @ $false
        @ ^ [Uu: b] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_132_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option2970150418924381261_c_nat] :
      ( ( Option != none_P8487441334512977628_c_nat )
      = ( case_o5420598178156702154_c_nat @ $false
        @ ^ [Uu: produc4862256710654508797_c_nat] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_133_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option6424319895471668383_c_nat] :
      ( ( Option = none_o1329286346417566254_c_nat )
      = ( case_o658134151661089564_c_nat @ $true
        @ ^ [Uu: option7520157102916957007_c_nat] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_134_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_c] :
      ( ( Option = none_c )
      = ( case_option_o_c @ $true
        @ ^ [Uu: c] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_135_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_nat] :
      ( ( Option = none_nat )
      = ( case_option_o_nat @ $true
        @ ^ [Uu: nat] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_136_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option7520157102916957007_c_nat] :
      ( ( Option = none_P6573378090780909534_c_nat )
      = ( case_o3625425862587997644_c_nat @ $true
        @ ^ [Uu: product_prod_c_nat] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_137_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_b] :
      ( ( Option = none_b )
      = ( case_option_o_b @ $true
        @ ^ [Uu: b] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_138_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option2970150418924381261_c_nat] :
      ( ( Option = none_P8487441334512977628_c_nat )
      = ( case_o5420598178156702154_c_nat @ $true
        @ ^ [Uu: produc4862256710654508797_c_nat] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_139_mem__case__prodE,axiom,
    ! [Z: b,C: c > nat > set_b,P: product_prod_c_nat] :
      ( ( member_b @ Z @ ( produc454361093202844089_set_b @ C @ P ) )
     => ~ ! [X: c,Y: nat] :
            ( ( P
              = ( product_Pair_c_nat @ X @ Y ) )
           => ~ ( member_b @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_140_mem__case__prodE,axiom,
    ! [Z: b,C: b > option7520157102916957007_c_nat > set_b,P: produc4862256710654508797_c_nat] :
      ( ( member_b @ Z @ ( produc4086673573166660539_set_b @ C @ P ) )
     => ~ ! [X: b,Y: option7520157102916957007_c_nat] :
            ( ( P
              = ( produc5716802255202478839_c_nat @ X @ Y ) )
           => ~ ( member_b @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_141_mem__case__prodE,axiom,
    ! [Z: nat,C: c > nat > set_nat,P: product_prod_c_nat] :
      ( ( member_nat @ Z @ ( produc4234106380533341996et_nat @ C @ P ) )
     => ~ ! [X: c,Y: nat] :
            ( ( P
              = ( product_Pair_c_nat @ X @ Y ) )
           => ~ ( member_nat @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_142_mem__case__prodE,axiom,
    ! [Z: c,C: c > nat > set_c,P: product_prod_c_nat] :
      ( ( member_c @ Z @ ( produc454361097506072890_set_c @ C @ P ) )
     => ~ ! [X: c,Y: nat] :
            ( ( P
              = ( product_Pair_c_nat @ X @ Y ) )
           => ~ ( member_c @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_143_mem__case__prodE,axiom,
    ! [Z: b,C: b > c > set_b,P: product_prod_b_c] :
      ( ( member_b @ Z @ ( produc3958908711191305138_set_b @ C @ P ) )
     => ~ ! [X: b,Y: c] :
            ( ( P
              = ( product_Pair_b_c @ X @ Y ) )
           => ~ ( member_b @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_144_mem__case__prodE,axiom,
    ! [Z: nat,C: b > c > set_nat,P: product_prod_b_c] :
      ( ( member_nat @ Z @ ( produc8716184571927494899et_nat @ C @ P ) )
     => ~ ! [X: b,Y: c] :
            ( ( P
              = ( product_Pair_b_c @ X @ Y ) )
           => ~ ( member_nat @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_145_mem__case__prodE,axiom,
    ! [Z: c,C: b > c > set_c,P: product_prod_b_c] :
      ( ( member_c @ Z @ ( produc3958908715494533939_set_c @ C @ P ) )
     => ~ ! [X: b,Y: c] :
            ( ( P
              = ( product_Pair_b_c @ X @ Y ) )
           => ~ ( member_c @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_146_mem__case__prodE,axiom,
    ! [Z: b,C: b > b > set_b,P: product_prod_b_b] :
      ( ( member_b @ Z @ ( produc2614443674381817203_set_b @ C @ P ) )
     => ~ ! [X: b,Y: b] :
            ( ( P
              = ( product_Pair_b_b @ X @ Y ) )
           => ~ ( member_b @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_147_mem__case__prodE,axiom,
    ! [Z: nat,C: b > b > set_nat,P: product_prod_b_b] :
      ( ( member_nat @ Z @ ( produc5893810021816043762et_nat @ C @ P ) )
     => ~ ! [X: b,Y: b] :
            ( ( P
              = ( product_Pair_b_b @ X @ Y ) )
           => ~ ( member_nat @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_148_mem__case__prodE,axiom,
    ! [Z: c,C: b > b > set_c,P: product_prod_b_b] :
      ( ( member_c @ Z @ ( produc2614443678685046004_set_c @ C @ P ) )
     => ~ ! [X: b,Y: b] :
            ( ( P
              = ( product_Pair_b_b @ X @ Y ) )
           => ~ ( member_c @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_149_option_Osimps_I4_J,axiom,
    ! [F1: $o,F2: product_prod_c_nat > $o] :
      ( ( case_o3625425862587997644_c_nat @ F1 @ F2 @ none_P6573378090780909534_c_nat )
      = F1 ) ).

% option.simps(4)
thf(fact_150_option_Osimps_I4_J,axiom,
    ! [F1: $o,F2: produc4862256710654508797_c_nat > $o] :
      ( ( case_o5420598178156702154_c_nat @ F1 @ F2 @ none_P8487441334512977628_c_nat )
      = F1 ) ).

% option.simps(4)
thf(fact_151_option_Osimps_I4_J,axiom,
    ! [F1: $o,F2: b > $o] :
      ( ( case_option_o_b @ F1 @ F2 @ none_b )
      = F1 ) ).

% option.simps(4)
thf(fact_152_option_Osimps_I4_J,axiom,
    ! [F1: nat,F2: nat > nat] :
      ( ( case_option_nat_nat @ F1 @ F2 @ none_nat )
      = F1 ) ).

% option.simps(4)
thf(fact_153_option_Osimps_I4_J,axiom,
    ! [F1: c,F2: c > c] :
      ( ( case_option_c_c @ F1 @ F2 @ none_c )
      = F1 ) ).

% option.simps(4)
thf(fact_154_option_Osimps_I4_J,axiom,
    ! [F1: produc4862256710654508797_c_nat,F2: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( case_o1602653449097480839_c_nat @ F1 @ F2 @ none_P8487441334512977628_c_nat )
      = F1 ) ).

% option.simps(4)
thf(fact_155_option_Osimps_I4_J,axiom,
    ! [F1: produc4862256710654508797_c_nat,F2: product_prod_c_nat > produc4862256710654508797_c_nat] :
      ( ( case_o6498799718091186953_c_nat @ F1 @ F2 @ none_P6573378090780909534_c_nat )
      = F1 ) ).

% option.simps(4)
thf(fact_156_case__prodE,axiom,
    ! [C: b > produc4862256710654508797_c_nat > $o,P: produc6689720525226317995_c_nat] :
      ( ( produc6484547981188837958_nat_o @ C @ P )
     => ~ ! [X: b,Y: produc4862256710654508797_c_nat] :
            ( ( P
              = ( produc8868828497325608613_c_nat @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_157_case__prodE,axiom,
    ! [C: b > mappin7688773591249790157_nat_b > $o,P: produc3246764621125188789_nat_b] :
      ( ( produc5419437825036861918at_b_o @ C @ P )
     => ~ ! [X: b,Y: mappin7688773591249790157_nat_b] :
            ( ( P
              = ( produc8656096559574149925_nat_b @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_158_case__prodE,axiom,
    ! [C: b > list_P8155366108056616461_c_nat > $o,P: produc6358558248090350139_c_nat] :
      ( ( produc9156697109305850166_nat_o @ C @ P )
     => ~ ! [X: b,Y: list_P8155366108056616461_c_nat] :
            ( ( P
              = ( produc385356081875622325_c_nat @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_159_case__prodE,axiom,
    ! [C: b > b > $o,P: product_prod_b_b] :
      ( ( produc4772849988445941548_b_b_o @ C @ P )
     => ~ ! [X: b,Y: b] :
            ( ( P
              = ( product_Pair_b_b @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_160_case__prodE,axiom,
    ! [C: b > c > $o,P: product_prod_b_c] :
      ( ( produc1873627588981547243_b_c_o @ C @ P )
     => ~ ! [X: b,Y: c] :
            ( ( P
              = ( product_Pair_b_c @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_161_case__prodE,axiom,
    ! [C: b > option7520157102916957007_c_nat > $o,P: produc4862256710654508797_c_nat] :
      ( ( produc2544960137906461044_nat_o @ C @ P )
     => ~ ! [X: b,Y: option7520157102916957007_c_nat] :
            ( ( P
              = ( produc5716802255202478839_c_nat @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_162_case__prodE,axiom,
    ! [C: c > nat > $o,P: product_prod_c_nat] :
      ( ( produc7133942929724870258_nat_o @ C @ P )
     => ~ ! [X: c,Y: nat] :
            ( ( P
              = ( product_Pair_c_nat @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_163_case__prodD,axiom,
    ! [F: b > produc4862256710654508797_c_nat > $o,A: b,B: produc4862256710654508797_c_nat] :
      ( ( produc6484547981188837958_nat_o @ F @ ( produc8868828497325608613_c_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_164_case__prodD,axiom,
    ! [F: b > mappin7688773591249790157_nat_b > $o,A: b,B: mappin7688773591249790157_nat_b] :
      ( ( produc5419437825036861918at_b_o @ F @ ( produc8656096559574149925_nat_b @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_165_case__prodD,axiom,
    ! [F: b > list_P8155366108056616461_c_nat > $o,A: b,B: list_P8155366108056616461_c_nat] :
      ( ( produc9156697109305850166_nat_o @ F @ ( produc385356081875622325_c_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_166_case__prodD,axiom,
    ! [F: b > b > $o,A: b,B: b] :
      ( ( produc4772849988445941548_b_b_o @ F @ ( product_Pair_b_b @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_167_case__prodD,axiom,
    ! [F: b > c > $o,A: b,B: c] :
      ( ( produc1873627588981547243_b_c_o @ F @ ( product_Pair_b_c @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_168_case__prodD,axiom,
    ! [F: b > option7520157102916957007_c_nat > $o,A: b,B: option7520157102916957007_c_nat] :
      ( ( produc2544960137906461044_nat_o @ F @ ( produc5716802255202478839_c_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_169_case__prodD,axiom,
    ! [F: c > nat > $o,A: c,B: nat] :
      ( ( produc7133942929724870258_nat_o @ F @ ( product_Pair_c_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_170_prod__induct3,axiom,
    ! [P2: produc6689720525226317995_c_nat > $o,X4: produc6689720525226317995_c_nat] :
      ( ! [A3: b,B3: b,C2: option7520157102916957007_c_nat] : ( P2 @ ( produc8868828497325608613_c_nat @ A3 @ ( produc5716802255202478839_c_nat @ B3 @ C2 ) ) )
     => ( P2 @ X4 ) ) ).

% prod_induct3
thf(fact_171_prod__cases3,axiom,
    ! [Y4: produc6689720525226317995_c_nat] :
      ~ ! [A3: b,B3: b,C2: option7520157102916957007_c_nat] :
          ( Y4
         != ( produc8868828497325608613_c_nat @ A3 @ ( produc5716802255202478839_c_nat @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_172_Pair__inject,axiom,
    ! [A: b,B: produc4862256710654508797_c_nat,A2: b,B2: produc4862256710654508797_c_nat] :
      ( ( ( produc8868828497325608613_c_nat @ A @ B )
        = ( produc8868828497325608613_c_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_173_Pair__inject,axiom,
    ! [A: b,B: mappin7688773591249790157_nat_b,A2: b,B2: mappin7688773591249790157_nat_b] :
      ( ( ( produc8656096559574149925_nat_b @ A @ B )
        = ( produc8656096559574149925_nat_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_174_Pair__inject,axiom,
    ! [A: b,B: list_P8155366108056616461_c_nat,A2: b,B2: list_P8155366108056616461_c_nat] :
      ( ( ( produc385356081875622325_c_nat @ A @ B )
        = ( produc385356081875622325_c_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_175_Pair__inject,axiom,
    ! [A: b,B: c,A2: b,B2: c] :
      ( ( ( product_Pair_b_c @ A @ B )
        = ( product_Pair_b_c @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_176_Pair__inject,axiom,
    ! [A: b,B: b,A2: b,B2: b] :
      ( ( ( product_Pair_b_b @ A @ B )
        = ( product_Pair_b_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_177_Pair__inject,axiom,
    ! [A: b,B: option7520157102916957007_c_nat,A2: b,B2: option7520157102916957007_c_nat] :
      ( ( ( produc5716802255202478839_c_nat @ A @ B )
        = ( produc5716802255202478839_c_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_178_Pair__inject,axiom,
    ! [A: c,B: nat,A2: c,B2: nat] :
      ( ( ( product_Pair_c_nat @ A @ B )
        = ( product_Pair_c_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_179_prod__cases,axiom,
    ! [P2: produc6689720525226317995_c_nat > $o,P: produc6689720525226317995_c_nat] :
      ( ! [A3: b,B3: produc4862256710654508797_c_nat] : ( P2 @ ( produc8868828497325608613_c_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_180_prod__cases,axiom,
    ! [P2: produc3246764621125188789_nat_b > $o,P: produc3246764621125188789_nat_b] :
      ( ! [A3: b,B3: mappin7688773591249790157_nat_b] : ( P2 @ ( produc8656096559574149925_nat_b @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_181_prod__cases,axiom,
    ! [P2: produc6358558248090350139_c_nat > $o,P: produc6358558248090350139_c_nat] :
      ( ! [A3: b,B3: list_P8155366108056616461_c_nat] : ( P2 @ ( produc385356081875622325_c_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_182_prod__cases,axiom,
    ! [P2: product_prod_b_c > $o,P: product_prod_b_c] :
      ( ! [A3: b,B3: c] : ( P2 @ ( product_Pair_b_c @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_183_prod__cases,axiom,
    ! [P2: product_prod_b_b > $o,P: product_prod_b_b] :
      ( ! [A3: b,B3: b] : ( P2 @ ( product_Pair_b_b @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_184_prod__cases,axiom,
    ! [P2: produc4862256710654508797_c_nat > $o,P: produc4862256710654508797_c_nat] :
      ( ! [A3: b,B3: option7520157102916957007_c_nat] : ( P2 @ ( produc5716802255202478839_c_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_185_prod__cases,axiom,
    ! [P2: product_prod_c_nat > $o,P: product_prod_c_nat] :
      ( ! [A3: c,B3: nat] : ( P2 @ ( product_Pair_c_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_186_surj__pair,axiom,
    ! [P: produc6689720525226317995_c_nat] :
    ? [X: b,Y: produc4862256710654508797_c_nat] :
      ( P
      = ( produc8868828497325608613_c_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_187_surj__pair,axiom,
    ! [P: produc3246764621125188789_nat_b] :
    ? [X: b,Y: mappin7688773591249790157_nat_b] :
      ( P
      = ( produc8656096559574149925_nat_b @ X @ Y ) ) ).

% surj_pair
thf(fact_188_surj__pair,axiom,
    ! [P: produc6358558248090350139_c_nat] :
    ? [X: b,Y: list_P8155366108056616461_c_nat] :
      ( P
      = ( produc385356081875622325_c_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_189_surj__pair,axiom,
    ! [P: product_prod_b_c] :
    ? [X: b,Y: c] :
      ( P
      = ( product_Pair_b_c @ X @ Y ) ) ).

% surj_pair
thf(fact_190_surj__pair,axiom,
    ! [P: product_prod_b_b] :
    ? [X: b,Y: b] :
      ( P
      = ( product_Pair_b_b @ X @ Y ) ) ).

% surj_pair
thf(fact_191_surj__pair,axiom,
    ! [P: produc4862256710654508797_c_nat] :
    ? [X: b,Y: option7520157102916957007_c_nat] :
      ( P
      = ( produc5716802255202478839_c_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_192_surj__pair,axiom,
    ! [P: product_prod_c_nat] :
    ? [X: c,Y: nat] :
      ( P
      = ( product_Pair_c_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_193_mem__Collect__eq,axiom,
    ! [A: option_b,P2: option_b > $o] :
      ( ( member_option_b @ A @ ( collect_option_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_194_mem__Collect__eq,axiom,
    ! [A: set_b,P2: set_b > $o] :
      ( ( member_set_b @ A @ ( collect_set_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_195_mem__Collect__eq,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_196_mem__Collect__eq,axiom,
    ! [A: c,P2: c > $o] :
      ( ( member_c @ A @ ( collect_c @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_197_mem__Collect__eq,axiom,
    ! [A: option2970150418924381261_c_nat,P2: option2970150418924381261_c_nat > $o] :
      ( ( member8863226656820029924_c_nat @ A @ ( collec5647424909086326562_c_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_198_mem__Collect__eq,axiom,
    ! [A: produc4862256710654508797_c_nat,P2: produc4862256710654508797_c_nat > $o] :
      ( ( member7562873241046315796_c_nat @ A @ ( collec4170641529751027026_c_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_199_mem__Collect__eq,axiom,
    ! [A: product_prod_b_c,P2: product_prod_b_c > $o] :
      ( ( member7862447941013992593od_b_c @ A @ ( collec548942224018234067od_b_c @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_200_mem__Collect__eq,axiom,
    ! [A: produc6689720525226317995_c_nat,P2: produc6689720525226317995_c_nat > $o] :
      ( ( member1522478204950355394_c_nat @ A @ ( collec8595031719707015680_c_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_201_mem__Collect__eq,axiom,
    ! [A: b,P2: b > $o] :
      ( ( member_b @ A @ ( collect_b @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_202_Collect__mem__eq,axiom,
    ! [A4: set_option_b] :
      ( ( collect_option_b
        @ ^ [X3: option_b] : ( member_option_b @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_203_Collect__mem__eq,axiom,
    ! [A4: set_set_b] :
      ( ( collect_set_b
        @ ^ [X3: set_b] : ( member_set_b @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_204_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_205_Collect__mem__eq,axiom,
    ! [A4: set_c] :
      ( ( collect_c
        @ ^ [X3: c] : ( member_c @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_206_Collect__mem__eq,axiom,
    ! [A4: set_op6736743094888940163_c_nat] :
      ( ( collec5647424909086326562_c_nat
        @ ^ [X3: option2970150418924381261_c_nat] : ( member8863226656820029924_c_nat @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_207_Collect__mem__eq,axiom,
    ! [A4: set_Pr8806432033423503795_c_nat] :
      ( ( collec4170641529751027026_c_nat
        @ ^ [X3: produc4862256710654508797_c_nat] : ( member7562873241046315796_c_nat @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_208_Collect__mem__eq,axiom,
    ! [A4: set_Product_prod_b_c] :
      ( ( collec548942224018234067od_b_c
        @ ^ [X3: product_prod_b_c] : ( member7862447941013992593od_b_c @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_209_Collect__mem__eq,axiom,
    ! [A4: set_Pr93524137074774369_c_nat] :
      ( ( collec8595031719707015680_c_nat
        @ ^ [X3: produc6689720525226317995_c_nat] : ( member1522478204950355394_c_nat @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_210_Collect__mem__eq,axiom,
    ! [A4: set_b] :
      ( ( collect_b
        @ ^ [X3: b] : ( member_b @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_211_Collect__cong,axiom,
    ! [P2: b > $o,Q2: b > $o] :
      ( ! [X: b] :
          ( ( P2 @ X )
          = ( Q2 @ X ) )
     => ( ( collect_b @ P2 )
        = ( collect_b @ Q2 ) ) ) ).

% Collect_cong
thf(fact_212_Collect__cong,axiom,
    ! [P2: option2970150418924381261_c_nat > $o,Q2: option2970150418924381261_c_nat > $o] :
      ( ! [X: option2970150418924381261_c_nat] :
          ( ( P2 @ X )
          = ( Q2 @ X ) )
     => ( ( collec5647424909086326562_c_nat @ P2 )
        = ( collec5647424909086326562_c_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_213_Collect__cong,axiom,
    ! [P2: produc4862256710654508797_c_nat > $o,Q2: produc4862256710654508797_c_nat > $o] :
      ( ! [X: produc4862256710654508797_c_nat] :
          ( ( P2 @ X )
          = ( Q2 @ X ) )
     => ( ( collec4170641529751027026_c_nat @ P2 )
        = ( collec4170641529751027026_c_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_214_Collect__cong,axiom,
    ! [P2: product_prod_b_c > $o,Q2: product_prod_b_c > $o] :
      ( ! [X: product_prod_b_c] :
          ( ( P2 @ X )
          = ( Q2 @ X ) )
     => ( ( collec548942224018234067od_b_c @ P2 )
        = ( collec548942224018234067od_b_c @ Q2 ) ) ) ).

% Collect_cong
thf(fact_215_Collect__cong,axiom,
    ! [P2: produc6689720525226317995_c_nat > $o,Q2: produc6689720525226317995_c_nat > $o] :
      ( ! [X: produc6689720525226317995_c_nat] :
          ( ( P2 @ X )
          = ( Q2 @ X ) )
     => ( ( collec8595031719707015680_c_nat @ P2 )
        = ( collec8595031719707015680_c_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_216_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc6689720525226317995_c_nat] :
      ~ ! [A3: b,B3: produc4862256710654508797_c_nat] :
          ( Y4
         != ( produc8868828497325608613_c_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_217_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc3246764621125188789_nat_b] :
      ~ ! [A3: b,B3: mappin7688773591249790157_nat_b] :
          ( Y4
         != ( produc8656096559574149925_nat_b @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_218_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc6358558248090350139_c_nat] :
      ~ ! [A3: b,B3: list_P8155366108056616461_c_nat] :
          ( Y4
         != ( produc385356081875622325_c_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_219_old_Oprod_Oexhaust,axiom,
    ! [Y4: product_prod_b_c] :
      ~ ! [A3: b,B3: c] :
          ( Y4
         != ( product_Pair_b_c @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_220_old_Oprod_Oexhaust,axiom,
    ! [Y4: product_prod_b_b] :
      ~ ! [A3: b,B3: b] :
          ( Y4
         != ( product_Pair_b_b @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_221_old_Oprod_Oexhaust,axiom,
    ! [Y4: produc4862256710654508797_c_nat] :
      ~ ! [A3: b,B3: option7520157102916957007_c_nat] :
          ( Y4
         != ( produc5716802255202478839_c_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_222_old_Oprod_Oexhaust,axiom,
    ! [Y4: product_prod_c_nat] :
      ~ ! [A3: c,B3: nat] :
          ( Y4
         != ( product_Pair_c_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_223_internal__case__prod__def,axiom,
    produc7943889084985077937_nat_b = produc9009988077629141723_nat_b ).

% internal_case_prod_def
thf(fact_224_internal__case__prod__def,axiom,
    produc1522750722963985571od_c_b = produc6717796504295817721od_c_b ).

% internal_case_prod_def
thf(fact_225_internal__case__prod__def,axiom,
    produc2953648338404074133_b_c_o = produc1873627588981547243_b_c_o ).

% internal_case_prod_def
thf(fact_226_internal__case__prod__def,axiom,
    produc3127621417668983676_b_c_b = produc8837194378803474130_b_c_b ).

% internal_case_prod_def
thf(fact_227_internal__case__prod__def,axiom,
    produc5915076999641659709_b_b_b = produc2401277923921374355_b_b_b ).

% internal_case_prod_def
thf(fact_228_internal__case__prod__def,axiom,
    produc3864834167062724555_c_nat = produc3722540595886809633_c_nat ).

% internal_case_prod_def
thf(fact_229_internal__case__prod__def,axiom,
    produc2955722632843901005_c_nat = produc3293230613241633187_c_nat ).

% internal_case_prod_def
thf(fact_230_pred__equals__eq2,axiom,
    ! [R: set_Pr93524137074774369_c_nat,S: set_Pr93524137074774369_c_nat] :
      ( ( ( ^ [X3: b,Y3: produc4862256710654508797_c_nat] : ( member1522478204950355394_c_nat @ ( produc8868828497325608613_c_nat @ X3 @ Y3 ) @ R ) )
        = ( ^ [X3: b,Y3: produc4862256710654508797_c_nat] : ( member1522478204950355394_c_nat @ ( produc8868828497325608613_c_nat @ X3 @ Y3 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_231_pred__equals__eq2,axiom,
    ! [R: set_Pr3745380843212982421_nat_b,S: set_Pr3745380843212982421_nat_b] :
      ( ( ( ^ [X3: b,Y3: mappin7688773591249790157_nat_b] : ( member8160214478689597918_nat_b @ ( produc8656096559574149925_nat_b @ X3 @ Y3 ) @ R ) )
        = ( ^ [X3: b,Y3: mappin7688773591249790157_nat_b] : ( member8160214478689597918_nat_b @ ( produc8656096559574149925_nat_b @ X3 @ Y3 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_232_pred__equals__eq2,axiom,
    ! [R: set_Pr55366283650883953_c_nat,S: set_Pr55366283650883953_c_nat] :
      ( ( ( ^ [X3: b,Y3: list_P8155366108056616461_c_nat] : ( member9033503127983833042_c_nat @ ( produc385356081875622325_c_nat @ X3 @ Y3 ) @ R ) )
        = ( ^ [X3: b,Y3: list_P8155366108056616461_c_nat] : ( member9033503127983833042_c_nat @ ( produc385356081875622325_c_nat @ X3 @ Y3 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_233_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_b_c,S: set_Product_prod_b_c] :
      ( ( ( ^ [X3: b,Y3: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ R ) )
        = ( ^ [X3: b,Y3: c] : ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X3 @ Y3 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_234_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
      ( ( ( ^ [X3: b,Y3: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R ) )
        = ( ^ [X3: b,Y3: b] : ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_235_pred__equals__eq2,axiom,
    ! [R: set_Pr8806432033423503795_c_nat,S: set_Pr8806432033423503795_c_nat] :
      ( ( ( ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X3 @ Y3 ) @ R ) )
        = ( ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X3 @ Y3 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_236_pred__equals__eq2,axiom,
    ! [R: set_Pr6903500605879609269_c_nat,S: set_Pr6903500605879609269_c_nat] :
      ( ( ( ^ [X3: c,Y3: nat] : ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X3 @ Y3 ) @ R ) )
        = ( ^ [X3: c,Y3: nat] : ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X3 @ Y3 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_237_internal__case__prod__conv,axiom,
    ! [C: c > nat > produc4862256710654508797_c_nat,A: c,B: nat] :
      ( ( produc2955722632843901005_c_nat @ C @ ( product_Pair_c_nat @ A @ B ) )
      = ( C @ A @ B ) ) ).

% internal_case_prod_conv
thf(fact_238_internal__case__prod__conv,axiom,
    ! [C: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,A: b,B: option7520157102916957007_c_nat] :
      ( ( produc3864834167062724555_c_nat @ C @ ( produc5716802255202478839_c_nat @ A @ B ) )
      = ( C @ A @ B ) ) ).

% internal_case_prod_conv
thf(fact_239_Mapping__lookup__None__intro,axiom,
    ! [X4: b,F: list_P7417839048565863355_c_nat] :
      ( ~ ( member_b @ X4 @ ( mmap_k3080892797436903101_c_nat @ F ) )
     => ( ( mmap_l5026999719965937591_c_nat @ F @ X4 )
        = none_P8487441334512977628_c_nat ) ) ).

% Mapping_lookup_None_intro
thf(fact_240_Mapping__lookup__None__intro,axiom,
    ! [X4: nat,F: list_P2993859830824179253_nat_c] :
      ( ~ ( member_nat @ X4 @ ( mmap_keys_nat_c @ F ) )
     => ( ( mmap_lookup_nat_c @ F @ X4 )
        = none_c ) ) ).

% Mapping_lookup_None_intro
thf(fact_241_Mapping__lookup__None__intro,axiom,
    ! [X4: c,F: list_P267710562049004973od_c_c] :
      ( ~ ( member_c @ X4 @ ( mmap_keys_c_c @ F ) )
     => ( ( mmap_lookup_c_c @ F @ X4 )
        = none_c ) ) ).

% Mapping_lookup_None_intro
thf(fact_242_Mapping__lookup__None__intro,axiom,
    ! [X4: b,F: list_P832325522607645677od_b_b] :
      ( ~ ( member_b @ X4 @ ( mmap_keys_b_b @ F ) )
     => ( ( mmap_lookup_b_b @ F @ X4 )
        = none_b ) ) ).

% Mapping_lookup_None_intro
thf(fact_243_Mapping__lookup__None__intro,axiom,
    ! [X4: nat,F: list_P2922825790777833268_nat_b] :
      ( ~ ( member_nat @ X4 @ ( mmap_keys_nat_b @ F ) )
     => ( ( mmap_lookup_nat_b @ F @ X4 )
        = none_b ) ) ).

% Mapping_lookup_None_intro
thf(fact_244_Mapping__lookup__None__intro,axiom,
    ! [X4: c,F: list_P196676522002658988od_c_b] :
      ( ~ ( member_c @ X4 @ ( mmap_keys_c_b @ F ) )
     => ( ( mmap_lookup_c_b @ F @ X4 )
        = none_b ) ) ).

% Mapping_lookup_None_intro
thf(fact_245_Mapping__lookup__None__intro,axiom,
    ! [X4: b,F: list_P9189103929441591694_b_nat] :
      ( ~ ( member_b @ X4 @ ( mmap_keys_b_nat @ F ) )
     => ( ( mmap_lookup_b_nat @ F @ X4 )
        = none_nat ) ) ).

% Mapping_lookup_None_intro
thf(fact_246_Mapping__lookup__None__intro,axiom,
    ! [X4: nat,F: list_P6011104703257516679at_nat] :
      ( ~ ( member_nat @ X4 @ ( mmap_keys_nat_nat @ F ) )
     => ( ( mmap_lookup_nat_nat @ F @ X4 )
        = none_nat ) ) ).

% Mapping_lookup_None_intro
thf(fact_247_Mapping__lookup__None__intro,axiom,
    ! [X4: b,F: list_P903359562653991662od_b_c] :
      ( ~ ( member_b @ X4 @ ( mmap_keys_b_c @ F ) )
     => ( ( mmap_lookup_b_c @ F @ X4 )
        = none_c ) ) ).

% Mapping_lookup_None_intro
thf(fact_248_Mapping__lookup__None__intro,axiom,
    ! [X4: c,F: list_P5561950507774946575_c_nat] :
      ( ~ ( member_c @ X4 @ ( mmap_keys_c_nat @ F ) )
     => ( ( mmap_lookup_c_nat @ F @ X4 )
        = none_nat ) ) ).

% Mapping_lookup_None_intro
thf(fact_249_Mapping__not__keys__intro,axiom,
    ! [F: list_P7417839048565863355_c_nat,X4: b] :
      ( ( ( mmap_l5026999719965937591_c_nat @ F @ X4 )
        = none_P8487441334512977628_c_nat )
     => ~ ( member_b @ X4 @ ( mmap_k3080892797436903101_c_nat @ F ) ) ) ).

% Mapping_not_keys_intro
thf(fact_250_Mapping__not__keys__intro,axiom,
    ! [F: list_P2993859830824179253_nat_c,X4: nat] :
      ( ( ( mmap_lookup_nat_c @ F @ X4 )
        = none_c )
     => ~ ( member_nat @ X4 @ ( mmap_keys_nat_c @ F ) ) ) ).

% Mapping_not_keys_intro
thf(fact_251_Mapping__not__keys__intro,axiom,
    ! [F: list_P267710562049004973od_c_c,X4: c] :
      ( ( ( mmap_lookup_c_c @ F @ X4 )
        = none_c )
     => ~ ( member_c @ X4 @ ( mmap_keys_c_c @ F ) ) ) ).

% Mapping_not_keys_intro
thf(fact_252_Mapping__not__keys__intro,axiom,
    ! [F: list_P832325522607645677od_b_b,X4: b] :
      ( ( ( mmap_lookup_b_b @ F @ X4 )
        = none_b )
     => ~ ( member_b @ X4 @ ( mmap_keys_b_b @ F ) ) ) ).

% Mapping_not_keys_intro
thf(fact_253_Mapping__not__keys__intro,axiom,
    ! [F: list_P2922825790777833268_nat_b,X4: nat] :
      ( ( ( mmap_lookup_nat_b @ F @ X4 )
        = none_b )
     => ~ ( member_nat @ X4 @ ( mmap_keys_nat_b @ F ) ) ) ).

% Mapping_not_keys_intro
thf(fact_254_Mapping__not__keys__intro,axiom,
    ! [F: list_P196676522002658988od_c_b,X4: c] :
      ( ( ( mmap_lookup_c_b @ F @ X4 )
        = none_b )
     => ~ ( member_c @ X4 @ ( mmap_keys_c_b @ F ) ) ) ).

% Mapping_not_keys_intro
thf(fact_255_Mapping__not__keys__intro,axiom,
    ! [F: list_P9189103929441591694_b_nat,X4: b] :
      ( ( ( mmap_lookup_b_nat @ F @ X4 )
        = none_nat )
     => ~ ( member_b @ X4 @ ( mmap_keys_b_nat @ F ) ) ) ).

% Mapping_not_keys_intro
thf(fact_256_Mapping__not__keys__intro,axiom,
    ! [F: list_P6011104703257516679at_nat,X4: nat] :
      ( ( ( mmap_lookup_nat_nat @ F @ X4 )
        = none_nat )
     => ~ ( member_nat @ X4 @ ( mmap_keys_nat_nat @ F ) ) ) ).

% Mapping_not_keys_intro
thf(fact_257_Mapping__not__keys__intro,axiom,
    ! [F: list_P903359562653991662od_b_c,X4: b] :
      ( ( ( mmap_lookup_b_c @ F @ X4 )
        = none_c )
     => ~ ( member_b @ X4 @ ( mmap_keys_b_c @ F ) ) ) ).

% Mapping_not_keys_intro
thf(fact_258_Mapping__not__keys__intro,axiom,
    ! [F: list_P5561950507774946575_c_nat,X4: c] :
      ( ( ( mmap_lookup_c_nat @ F @ X4 )
        = none_nat )
     => ~ ( member_c @ X4 @ ( mmap_keys_c_nat @ F ) ) ) ).

% Mapping_not_keys_intro
thf(fact_259_Mapping__keys__intro,axiom,
    ! [F: list_P7417839048565863355_c_nat,X4: b] :
      ( ( ( mmap_l5026999719965937591_c_nat @ F @ X4 )
       != none_P8487441334512977628_c_nat )
     => ( member_b @ X4 @ ( mmap_k3080892797436903101_c_nat @ F ) ) ) ).

% Mapping_keys_intro
thf(fact_260_Mapping__keys__intro,axiom,
    ! [F: list_P2993859830824179253_nat_c,X4: nat] :
      ( ( ( mmap_lookup_nat_c @ F @ X4 )
       != none_c )
     => ( member_nat @ X4 @ ( mmap_keys_nat_c @ F ) ) ) ).

% Mapping_keys_intro
thf(fact_261_Mapping__keys__intro,axiom,
    ! [F: list_P267710562049004973od_c_c,X4: c] :
      ( ( ( mmap_lookup_c_c @ F @ X4 )
       != none_c )
     => ( member_c @ X4 @ ( mmap_keys_c_c @ F ) ) ) ).

% Mapping_keys_intro
thf(fact_262_Mapping__keys__intro,axiom,
    ! [F: list_P832325522607645677od_b_b,X4: b] :
      ( ( ( mmap_lookup_b_b @ F @ X4 )
       != none_b )
     => ( member_b @ X4 @ ( mmap_keys_b_b @ F ) ) ) ).

% Mapping_keys_intro
thf(fact_263_Mapping__keys__intro,axiom,
    ! [F: list_P2922825790777833268_nat_b,X4: nat] :
      ( ( ( mmap_lookup_nat_b @ F @ X4 )
       != none_b )
     => ( member_nat @ X4 @ ( mmap_keys_nat_b @ F ) ) ) ).

% Mapping_keys_intro
thf(fact_264_Mapping__keys__intro,axiom,
    ! [F: list_P196676522002658988od_c_b,X4: c] :
      ( ( ( mmap_lookup_c_b @ F @ X4 )
       != none_b )
     => ( member_c @ X4 @ ( mmap_keys_c_b @ F ) ) ) ).

% Mapping_keys_intro
thf(fact_265_Mapping__keys__intro,axiom,
    ! [F: list_P9189103929441591694_b_nat,X4: b] :
      ( ( ( mmap_lookup_b_nat @ F @ X4 )
       != none_nat )
     => ( member_b @ X4 @ ( mmap_keys_b_nat @ F ) ) ) ).

% Mapping_keys_intro
thf(fact_266_Mapping__keys__intro,axiom,
    ! [F: list_P6011104703257516679at_nat,X4: nat] :
      ( ( ( mmap_lookup_nat_nat @ F @ X4 )
       != none_nat )
     => ( member_nat @ X4 @ ( mmap_keys_nat_nat @ F ) ) ) ).

% Mapping_keys_intro
thf(fact_267_Mapping__keys__intro,axiom,
    ! [F: list_P903359562653991662od_b_c,X4: b] :
      ( ( ( mmap_lookup_b_c @ F @ X4 )
       != none_c )
     => ( member_b @ X4 @ ( mmap_keys_b_c @ F ) ) ) ).

% Mapping_keys_intro
thf(fact_268_Mapping__keys__intro,axiom,
    ! [F: list_P5561950507774946575_c_nat,X4: c] :
      ( ( ( mmap_lookup_c_nat @ F @ X4 )
       != none_nat )
     => ( member_c @ X4 @ ( mmap_keys_c_nat @ F ) ) ) ).

% Mapping_keys_intro
thf(fact_269_curry__case__prod,axiom,
    ! [F: c > nat > $o] :
      ( ( produc8799735574653552657_nat_o @ ( produc7133942929724870258_nat_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_270_curry__case__prod,axiom,
    ! [F: b > option7520157102916957007_c_nat > $o] :
      ( ( produc7520937516567986579_nat_o @ ( produc2544960137906461044_nat_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_271_curry__case__prod,axiom,
    ! [F: b > option7520157102916957007_c_nat > b] :
      ( ( produc1575552564918139770_nat_b @ ( produc9009988077629141723_nat_b @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_272_curry__case__prod,axiom,
    ! [F: b > c > product_prod_c_b] :
      ( ( produc1480702574582671514od_c_b @ ( produc6717796504295817721od_c_b @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_273_curry__case__prod,axiom,
    ! [F: b > c > $o] :
      ( ( product_curry_b_c_o @ ( produc1873627588981547243_b_c_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_274_curry__case__prod,axiom,
    ! [F: b > c > b] :
      ( ( product_curry_b_c_b @ ( produc8837194378803474130_b_c_b @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_275_curry__case__prod,axiom,
    ! [F: b > b > b] :
      ( ( product_curry_b_b_b @ ( produc2401277923921374355_b_b_b @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_276_curry__case__prod,axiom,
    ! [F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat] :
      ( ( produc5695290940949948482_c_nat @ ( produc3722540595886809633_c_nat @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_277_curry__case__prod,axiom,
    ! [F: c > nat > produc4862256710654508797_c_nat] :
      ( ( produc2131212485941128772_c_nat @ ( produc3293230613241633187_c_nat @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_278_case__prod__curry,axiom,
    ! [F: product_prod_c_nat > $o] :
      ( ( produc7133942929724870258_nat_o @ ( produc8799735574653552657_nat_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_279_case__prod__curry,axiom,
    ! [F: produc4862256710654508797_c_nat > $o] :
      ( ( produc2544960137906461044_nat_o @ ( produc7520937516567986579_nat_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_280_case__prod__curry,axiom,
    ! [F: produc4862256710654508797_c_nat > b] :
      ( ( produc9009988077629141723_nat_b @ ( produc1575552564918139770_nat_b @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_281_case__prod__curry,axiom,
    ! [F: product_prod_b_c > product_prod_c_b] :
      ( ( produc6717796504295817721od_c_b @ ( produc1480702574582671514od_c_b @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_282_case__prod__curry,axiom,
    ! [F: product_prod_b_c > $o] :
      ( ( produc1873627588981547243_b_c_o @ ( product_curry_b_c_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_283_case__prod__curry,axiom,
    ! [F: product_prod_b_c > b] :
      ( ( produc8837194378803474130_b_c_b @ ( product_curry_b_c_b @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_284_case__prod__curry,axiom,
    ! [F: product_prod_b_b > b] :
      ( ( produc2401277923921374355_b_b_b @ ( product_curry_b_b_b @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_285_case__prod__curry,axiom,
    ! [F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( produc3722540595886809633_c_nat @ ( produc5695290940949948482_c_nat @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_286_case__prod__curry,axiom,
    ! [F: product_prod_c_nat > produc4862256710654508797_c_nat] :
      ( ( produc3293230613241633187_c_nat @ ( produc2131212485941128772_c_nat @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_287_curryI,axiom,
    ! [F: produc6689720525226317995_c_nat > $o,A: b,B: produc4862256710654508797_c_nat] :
      ( ( F @ ( produc8868828497325608613_c_nat @ A @ B ) )
     => ( produc6983985203698738021_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_288_curryI,axiom,
    ! [F: produc3246764621125188789_nat_b > $o,A: b,B: mappin7688773591249790157_nat_b] :
      ( ( F @ ( produc8656096559574149925_nat_b @ A @ B ) )
     => ( produc3606654669749784831at_b_o @ F @ A @ B ) ) ).

% curryI
thf(fact_289_curryI,axiom,
    ! [F: produc6358558248090350139_c_nat > $o,A: b,B: list_P8155366108056616461_c_nat] :
      ( ( F @ ( produc385356081875622325_c_nat @ A @ B ) )
     => ( produc7188371016155025109_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_290_curryI,axiom,
    ! [F: product_prod_b_c > $o,A: b,B: c] :
      ( ( F @ ( product_Pair_b_c @ A @ B ) )
     => ( product_curry_b_c_o @ F @ A @ B ) ) ).

% curryI
thf(fact_291_curryI,axiom,
    ! [F: product_prod_b_b > $o,A: b,B: b] :
      ( ( F @ ( product_Pair_b_b @ A @ B ) )
     => ( product_curry_b_b_o @ F @ A @ B ) ) ).

% curryI
thf(fact_292_curryI,axiom,
    ! [F: produc4862256710654508797_c_nat > $o,A: b,B: option7520157102916957007_c_nat] :
      ( ( F @ ( produc5716802255202478839_c_nat @ A @ B ) )
     => ( produc7520937516567986579_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_293_curryI,axiom,
    ! [F: product_prod_c_nat > $o,A: c,B: nat] :
      ( ( F @ ( product_Pair_c_nat @ A @ B ) )
     => ( produc8799735574653552657_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_294_split__part,axiom,
    ! [P2: $o,Q2: b > c > $o] :
      ( ( produc1873627588981547243_b_c_o
        @ ^ [A5: b,B4: c] :
            ( P2
            & ( Q2 @ A5 @ B4 ) ) )
      = ( ^ [Ab: product_prod_b_c] :
            ( P2
            & ( produc1873627588981547243_b_c_o @ Q2 @ Ab ) ) ) ) ).

% split_part
thf(fact_295_curry__conv,axiom,
    ( produc2131212485941128772_c_nat
    = ( ^ [F3: product_prod_c_nat > produc4862256710654508797_c_nat,A5: c,B4: nat] : ( F3 @ ( product_Pair_c_nat @ A5 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_296_curry__conv,axiom,
    ( produc5695290940949948482_c_nat
    = ( ^ [F3: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,A5: b,B4: option7520157102916957007_c_nat] : ( F3 @ ( produc5716802255202478839_c_nat @ A5 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_297_curry__conv,axiom,
    ( produc8799735574653552657_nat_o
    = ( ^ [F3: product_prod_c_nat > $o,A5: c,B4: nat] : ( F3 @ ( product_Pair_c_nat @ A5 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_298_curry__conv,axiom,
    ( produc7520937516567986579_nat_o
    = ( ^ [F3: produc4862256710654508797_c_nat > $o,A5: b,B4: option7520157102916957007_c_nat] : ( F3 @ ( produc5716802255202478839_c_nat @ A5 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_299_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_b_c] :
      ( produc1873627588981547243_b_c_o
      @ ^ [Uu: b,Uv: c] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_300_curry__K,axiom,
    ! [C: produc4862256710654508797_c_nat] :
      ( ( produc2131212485941128772_c_nat
        @ ^ [X3: product_prod_c_nat] : C )
      = ( ^ [X3: c,Y3: nat] : C ) ) ).

% curry_K
thf(fact_301_curry__K,axiom,
    ! [C: produc4862256710654508797_c_nat] :
      ( ( produc5695290940949948482_c_nat
        @ ^ [X3: produc4862256710654508797_c_nat] : C )
      = ( ^ [X3: b,Y3: option7520157102916957007_c_nat] : C ) ) ).

% curry_K
thf(fact_302_curry__K,axiom,
    ! [C: $o] :
      ( ( produc8799735574653552657_nat_o
        @ ^ [X3: product_prod_c_nat] : C )
      = ( ^ [X3: c,Y3: nat] : C ) ) ).

% curry_K
thf(fact_303_curry__K,axiom,
    ! [C: $o] :
      ( ( produc7520937516567986579_nat_o
        @ ^ [X3: produc4862256710654508797_c_nat] : C )
      = ( ^ [X3: b,Y3: option7520157102916957007_c_nat] : C ) ) ).

% curry_K
thf(fact_304_curryD,axiom,
    ! [F: produc6689720525226317995_c_nat > $o,A: b,B: produc4862256710654508797_c_nat] :
      ( ( produc6983985203698738021_nat_o @ F @ A @ B )
     => ( F @ ( produc8868828497325608613_c_nat @ A @ B ) ) ) ).

% curryD
thf(fact_305_curryD,axiom,
    ! [F: produc3246764621125188789_nat_b > $o,A: b,B: mappin7688773591249790157_nat_b] :
      ( ( produc3606654669749784831at_b_o @ F @ A @ B )
     => ( F @ ( produc8656096559574149925_nat_b @ A @ B ) ) ) ).

% curryD
thf(fact_306_curryD,axiom,
    ! [F: produc6358558248090350139_c_nat > $o,A: b,B: list_P8155366108056616461_c_nat] :
      ( ( produc7188371016155025109_nat_o @ F @ A @ B )
     => ( F @ ( produc385356081875622325_c_nat @ A @ B ) ) ) ).

% curryD
thf(fact_307_curryD,axiom,
    ! [F: product_prod_b_c > $o,A: b,B: c] :
      ( ( product_curry_b_c_o @ F @ A @ B )
     => ( F @ ( product_Pair_b_c @ A @ B ) ) ) ).

% curryD
thf(fact_308_curryD,axiom,
    ! [F: product_prod_b_b > $o,A: b,B: b] :
      ( ( product_curry_b_b_o @ F @ A @ B )
     => ( F @ ( product_Pair_b_b @ A @ B ) ) ) ).

% curryD
thf(fact_309_curryD,axiom,
    ! [F: produc4862256710654508797_c_nat > $o,A: b,B: option7520157102916957007_c_nat] :
      ( ( produc7520937516567986579_nat_o @ F @ A @ B )
     => ( F @ ( produc5716802255202478839_c_nat @ A @ B ) ) ) ).

% curryD
thf(fact_310_curryD,axiom,
    ! [F: product_prod_c_nat > $o,A: c,B: nat] :
      ( ( produc8799735574653552657_nat_o @ F @ A @ B )
     => ( F @ ( product_Pair_c_nat @ A @ B ) ) ) ).

% curryD
thf(fact_311_curryE,axiom,
    ! [F: product_prod_b_c > $o,A: b,B: c] :
      ( ( product_curry_b_c_o @ F @ A @ B )
     => ( F @ ( product_Pair_b_c @ A @ B ) ) ) ).

% curryE
thf(fact_312_curryE,axiom,
    ! [F: product_prod_b_b > $o,A: b,B: b] :
      ( ( product_curry_b_b_o @ F @ A @ B )
     => ( F @ ( product_Pair_b_b @ A @ B ) ) ) ).

% curryE
thf(fact_313_curryE,axiom,
    ! [F: produc4862256710654508797_c_nat > $o,A: b,B: option7520157102916957007_c_nat] :
      ( ( produc7520937516567986579_nat_o @ F @ A @ B )
     => ( F @ ( produc5716802255202478839_c_nat @ A @ B ) ) ) ).

% curryE
thf(fact_314_curryE,axiom,
    ! [F: product_prod_c_nat > $o,A: c,B: nat] :
      ( ( produc8799735574653552657_nat_o @ F @ A @ B )
     => ( F @ ( product_Pair_c_nat @ A @ B ) ) ) ).

% curryE
thf(fact_315_init__in__keys__s,axiom,
    member_b @ init @ ( mmap_k3080892797436903101_c_nat @ s3 ) ).

% init_in_keys_s
thf(fact_316_drop__cur__def,axiom,
    ( drop_cur_b_c
    = ( ^ [I: nat] :
          ( produc3722540595886809633_c_nat
          @ ^ [Q: b,Tstp2: option7520157102916957007_c_nat] :
              ( produc5716802255202478839_c_nat @ Q
              @ ( case_o8358378205851023707_c_nat @ Tstp2
                @ ( produc7737764547318506485_c_nat
                  @ ^ [Ts: c,Tp: nat] : ( if_opt8655011569862983689_c_nat @ ( Tp = I ) @ none_P6573378090780909534_c_nat @ Tstp2 ) )
                @ Tstp2 ) ) ) ) ) ).

% drop_cur_def
thf(fact_317_mmap__map__keys,axiom,
    ! [F: b > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,M: list_P7417839048565863355_c_nat] :
      ( ( mmap_k3080892797436903101_c_nat @ ( mmap_m881505119971688169_c_nat @ F @ M ) )
      = ( mmap_k3080892797436903101_c_nat @ M ) ) ).

% mmap_map_keys
thf(fact_318_The__split__eq,axiom,
    ! [X4: b,Y4: option7520157102916957007_c_nat] :
      ( ( the_Pr4418202073612014518_c_nat
        @ ( produc2544960137906461044_nat_o
          @ ^ [X5: b,Y5: option7520157102916957007_c_nat] :
              ( ( X4 = X5 )
              & ( Y4 = Y5 ) ) ) )
      = ( produc5716802255202478839_c_nat @ X4 @ Y4 ) ) ).

% The_split_eq
thf(fact_319_The__split__eq,axiom,
    ! [X4: c,Y4: nat] :
      ( ( the_Pr1992744297546409400_c_nat
        @ ( produc7133942929724870258_nat_o
          @ ^ [X5: c,Y5: nat] :
              ( ( X4 = X5 )
              & ( Y4 = Y5 ) ) ) )
      = ( product_Pair_c_nat @ X4 @ Y4 ) ) ).

% The_split_eq
thf(fact_320_plus__prod_Ocases,axiom,
    ! [X4: produc5740701590023533791_c_nat] :
      ~ ! [A3: c,B3: nat,C2: c,D: nat] :
          ( X4
         != ( produc3560461569812927383_c_nat @ ( product_Pair_c_nat @ A3 @ B3 ) @ ( product_Pair_c_nat @ C2 @ D ) ) ) ).

% plus_prod.cases
thf(fact_321_ssubst__Pair__rhs,axiom,
    ! [R2: b,S2: option7520157102916957007_c_nat,R: set_Pr8806432033423503795_c_nat,S3: option7520157102916957007_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ R2 @ S2 ) @ R )
     => ( ( S3 = S2 )
       => ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ R2 @ S3 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_322_ssubst__Pair__rhs,axiom,
    ! [R2: c,S2: nat,R: set_Pr6903500605879609269_c_nat,S3: nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ R2 @ S2 ) @ R )
     => ( ( S3 = S2 )
       => ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ R2 @ S3 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_323_same__fstI,axiom,
    ! [P2: b > $o,X4: b,Y6: option7520157102916957007_c_nat,Y4: option7520157102916957007_c_nat,R: b > set_Pr4914646408830571231_c_nat] :
      ( ( P2 @ X4 )
     => ( ( member6296142003466748712_c_nat @ ( produc4206217144015867191_c_nat @ Y6 @ Y4 ) @ ( R @ X4 ) )
       => ( member16691529365684868_c_nat @ ( produc6399785788075897747_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y6 ) @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) ) @ ( same_f4655507504445923682_c_nat @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_324_same__fstI,axiom,
    ! [P2: c > $o,X4: c,Y6: nat,Y4: nat,R: c > set_Pr1261947904930325089at_nat] :
      ( ( P2 @ X4 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Y4 ) @ ( R @ X4 ) )
       => ( member871170437899409800_c_nat @ ( produc3560461569812927383_c_nat @ ( product_Pair_c_nat @ X4 @ Y6 ) @ ( product_Pair_c_nat @ X4 @ Y4 ) ) @ ( same_fst_c_nat @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_325_in__lex__prod,axiom,
    ! [A: b,B: option7520157102916957007_c_nat,A2: b,B2: option7520157102916957007_c_nat,R2: set_Product_prod_b_b,S2: set_Pr4914646408830571231_c_nat] :
      ( ( member16691529365684868_c_nat @ ( produc6399785788075897747_c_nat @ ( produc5716802255202478839_c_nat @ A @ B ) @ ( produc5716802255202478839_c_nat @ A2 @ B2 ) ) @ ( lex_pr4265472524319399135_c_nat @ R2 @ S2 ) )
      = ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ A2 ) @ R2 )
        | ( ( A = A2 )
          & ( member6296142003466748712_c_nat @ ( produc4206217144015867191_c_nat @ B @ B2 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_326_in__lex__prod,axiom,
    ! [A: c,B: nat,A2: c,B2: nat,R2: set_Product_prod_c_c,S2: set_Pr1261947904930325089at_nat] :
      ( ( member871170437899409800_c_nat @ ( produc3560461569812927383_c_nat @ ( product_Pair_c_nat @ A @ B ) @ ( product_Pair_c_nat @ A2 @ B2 ) ) @ ( lex_prod_c_nat @ R2 @ S2 ) )
      = ( ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ A @ A2 ) @ R2 )
        | ( ( A = A2 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_327_mmap__map__def,axiom,
    ( mmap_m4119996586642187405_c_nat
    = ( ^ [F3: b > option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
          ( map_Pr5967270692098401216_c_nat
          @ ( produc3722540595886809633_c_nat
            @ ^ [K: b,V: option7520157102916957007_c_nat] : ( produc5716802255202478839_c_nat @ K @ ( F3 @ K @ V ) ) ) ) ) ) ).

% mmap_map_def
thf(fact_328_case__swap,axiom,
    ! [F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,P: produc4862256710654508797_c_nat] :
      ( ( produc2411843245718984257_c_nat
        @ ^ [Y3: option7520157102916957007_c_nat,X3: b] : ( F @ X3 @ Y3 )
        @ ( produc8310569085468513623_c_nat @ P ) )
      = ( produc3722540595886809633_c_nat @ F @ P ) ) ).

% case_swap
thf(fact_329_case__swap,axiom,
    ! [F: c > nat > produc4862256710654508797_c_nat,P: product_prod_c_nat] :
      ( ( produc6856352468954913281_c_nat
        @ ^ [Y3: nat,X3: c] : ( F @ X3 @ Y3 )
        @ ( product_swap_c_nat @ P ) )
      = ( produc3293230613241633187_c_nat @ F @ P ) ) ).

% case_swap
thf(fact_330_case__swap,axiom,
    ! [F: option7520157102916957007_c_nat > b > produc4862256710654508797_c_nat,P: produc7545861912598698981_nat_b] :
      ( ( produc3722540595886809633_c_nat
        @ ^ [Y3: b,X3: option7520157102916957007_c_nat] : ( F @ X3 @ Y3 )
        @ ( produc3216575356166240631_nat_b @ P ) )
      = ( produc2411843245718984257_c_nat @ F @ P ) ) ).

% case_swap
thf(fact_331_case__swap,axiom,
    ! [F: nat > c > produc4862256710654508797_c_nat,P: product_prod_nat_c] :
      ( ( produc3293230613241633187_c_nat
        @ ^ [Y3: c,X3: nat] : ( F @ X3 @ Y3 )
        @ ( product_swap_nat_c @ P ) )
      = ( produc6856352468954913281_c_nat @ F @ P ) ) ).

% case_swap
thf(fact_332_pair__imageI,axiom,
    ! [A: b,B: option7520157102916957007_c_nat,A4: set_Pr8806432033423503795_c_nat,F: b > option7520157102916957007_c_nat > b] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A @ B ) @ A4 )
     => ( member_b @ ( F @ A @ B ) @ ( image_2722527542522727419_nat_b @ ( produc9009988077629141723_nat_b @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_333_pair__imageI,axiom,
    ! [A: c,B: nat,A4: set_Pr6903500605879609269_c_nat,F: c > nat > b] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A @ B ) @ A4 )
     => ( member_b @ ( F @ A @ B ) @ ( image_124767200456794617_nat_b @ ( produc1350122042873596121_nat_b @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_334_pair__imageI,axiom,
    ! [A: b,B: option7520157102916957007_c_nat,A4: set_Pr8806432033423503795_c_nat,F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A @ B ) @ A4 )
     => ( member7562873241046315796_c_nat @ ( F @ A @ B ) @ ( image_7713823756821623041_c_nat @ ( produc3722540595886809633_c_nat @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_335_pair__imageI,axiom,
    ! [A: c,B: nat,A4: set_Pr6903500605879609269_c_nat,F: c > nat > produc4862256710654508797_c_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A @ B ) @ A4 )
     => ( member7562873241046315796_c_nat @ ( F @ A @ B ) @ ( image_1176608181733484675_c_nat @ ( produc3293230613241633187_c_nat @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_336_map__values__lookup,axiom,
    ! [F: b > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,M: list_P7417839048565863355_c_nat,Z: b,V2: produc4862256710654508797_c_nat] :
      ( ( ( mmap_l5026999719965937591_c_nat @ ( mmap_m881505119971688169_c_nat @ F @ M ) @ Z )
        = ( some_P2720002978653898840_c_nat @ V2 ) )
     => ? [V3: produc4862256710654508797_c_nat] :
          ( ( ( mmap_l5026999719965937591_c_nat @ M @ Z )
            = ( some_P2720002978653898840_c_nat @ V3 ) )
          & ( V2
            = ( F @ Z @ V3 ) ) ) ) ).

% map_values_lookup
thf(fact_337_not__None__eq,axiom,
    ! [X4: option2970150418924381261_c_nat] :
      ( ( X4 != none_P8487441334512977628_c_nat )
      = ( ? [Y3: produc4862256710654508797_c_nat] :
            ( X4
            = ( some_P2720002978653898840_c_nat @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_338_not__Some__eq,axiom,
    ! [X4: option2970150418924381261_c_nat] :
      ( ( ! [Y3: produc4862256710654508797_c_nat] :
            ( X4
           != ( some_P2720002978653898840_c_nat @ Y3 ) ) )
      = ( X4 = none_P8487441334512977628_c_nat ) ) ).

% not_Some_eq
thf(fact_339_pair__in__swap__image,axiom,
    ! [Y4: option7520157102916957007_c_nat,X4: b,A4: set_Pr8806432033423503795_c_nat] :
      ( ( member1023106406135730172_nat_b @ ( produc622808525900205847_nat_b @ Y4 @ X4 ) @ ( image_1174056921911037417_nat_b @ produc8310569085468513623_c_nat @ A4 ) )
      = ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_340_pair__in__swap__image,axiom,
    ! [Y4: nat,X4: c,A4: set_Pr6903500605879609269_c_nat] :
      ( ( member8962352060716553276_nat_c @ ( product_Pair_nat_c @ Y4 @ X4 ) @ ( image_1786354358617006891_nat_c @ product_swap_c_nat @ A4 ) )
      = ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X4 @ Y4 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_341_pair__in__swap__image,axiom,
    ! [Y4: b,X4: option7520157102916957007_c_nat,A4: set_Pr1066409585067842203_nat_b] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ Y4 @ X4 ) @ ( image_3433634698358497769_c_nat @ produc3216575356166240631_nat_b @ A4 ) )
      = ( member1023106406135730172_nat_b @ ( produc622808525900205847_nat_b @ X4 @ Y4 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_342_pair__in__swap__image,axiom,
    ! [Y4: c,X4: nat,A4: set_Pr4335409928928841947_nat_c] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ Y4 @ X4 ) @ ( image_6275841512968140075_c_nat @ product_swap_nat_c @ A4 ) )
      = ( member8962352060716553276_nat_c @ ( product_Pair_nat_c @ X4 @ Y4 ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_343_swap__simp,axiom,
    ! [X4: option7520157102916957007_c_nat,Y4: b] :
      ( ( produc3216575356166240631_nat_b @ ( produc622808525900205847_nat_b @ X4 @ Y4 ) )
      = ( produc5716802255202478839_c_nat @ Y4 @ X4 ) ) ).

% swap_simp
thf(fact_344_swap__simp,axiom,
    ! [X4: nat,Y4: c] :
      ( ( product_swap_nat_c @ ( product_Pair_nat_c @ X4 @ Y4 ) )
      = ( product_Pair_c_nat @ Y4 @ X4 ) ) ).

% swap_simp
thf(fact_345_swap__simp,axiom,
    ! [X4: b,Y4: option7520157102916957007_c_nat] :
      ( ( produc8310569085468513623_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) )
      = ( produc622808525900205847_nat_b @ Y4 @ X4 ) ) ).

% swap_simp
thf(fact_346_swap__simp,axiom,
    ! [X4: c,Y4: nat] :
      ( ( product_swap_c_nat @ ( product_Pair_c_nat @ X4 @ Y4 ) )
      = ( product_Pair_nat_c @ Y4 @ X4 ) ) ).

% swap_simp
thf(fact_347_option_Ocollapse,axiom,
    ! [Option: option2970150418924381261_c_nat] :
      ( ( Option != none_P8487441334512977628_c_nat )
     => ( ( some_P2720002978653898840_c_nat @ ( the_Pr3263588667584626537_c_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_348_option_Osel,axiom,
    ! [X2: produc4862256710654508797_c_nat] :
      ( ( the_Pr3263588667584626537_c_nat @ ( some_P2720002978653898840_c_nat @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_349_option_Oexhaust__sel,axiom,
    ! [Option: option2970150418924381261_c_nat] :
      ( ( Option != none_P8487441334512977628_c_nat )
     => ( Option
        = ( some_P2720002978653898840_c_nat @ ( the_Pr3263588667584626537_c_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_350_option_Osplit__sel,axiom,
    ! [P2: produc4862256710654508797_c_nat > $o,F1: produc4862256710654508797_c_nat,F2: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Option: option2970150418924381261_c_nat] :
      ( ( P2 @ ( case_o1602653449097480839_c_nat @ F1 @ F2 @ Option ) )
      = ( ( ( Option = none_P8487441334512977628_c_nat )
         => ( P2 @ F1 ) )
        & ( ( Option
            = ( some_P2720002978653898840_c_nat @ ( the_Pr3263588667584626537_c_nat @ Option ) ) )
         => ( P2 @ ( F2 @ ( the_Pr3263588667584626537_c_nat @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_351_option_Osplit__sel,axiom,
    ! [P2: produc4862256710654508797_c_nat > $o,F1: produc4862256710654508797_c_nat,F2: product_prod_c_nat > produc4862256710654508797_c_nat,Option: option7520157102916957007_c_nat] :
      ( ( P2 @ ( case_o6498799718091186953_c_nat @ F1 @ F2 @ Option ) )
      = ( ( ( Option = none_P6573378090780909534_c_nat )
         => ( P2 @ F1 ) )
        & ( ( Option
            = ( some_P8722241760384591706_c_nat @ ( the_Pr7350365623720035563_c_nat @ Option ) ) )
         => ( P2 @ ( F2 @ ( the_Pr7350365623720035563_c_nat @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_352_option_Osplit__sel__asm,axiom,
    ! [P2: produc4862256710654508797_c_nat > $o,F1: produc4862256710654508797_c_nat,F2: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Option: option2970150418924381261_c_nat] :
      ( ( P2 @ ( case_o1602653449097480839_c_nat @ F1 @ F2 @ Option ) )
      = ( ~ ( ( ( Option = none_P8487441334512977628_c_nat )
              & ~ ( P2 @ F1 ) )
            | ( ( Option
                = ( some_P2720002978653898840_c_nat @ ( the_Pr3263588667584626537_c_nat @ Option ) ) )
              & ~ ( P2 @ ( F2 @ ( the_Pr3263588667584626537_c_nat @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_353_option_Osplit__sel__asm,axiom,
    ! [P2: produc4862256710654508797_c_nat > $o,F1: produc4862256710654508797_c_nat,F2: product_prod_c_nat > produc4862256710654508797_c_nat,Option: option7520157102916957007_c_nat] :
      ( ( P2 @ ( case_o6498799718091186953_c_nat @ F1 @ F2 @ Option ) )
      = ( ~ ( ( ( Option = none_P6573378090780909534_c_nat )
              & ~ ( P2 @ F1 ) )
            | ( ( Option
                = ( some_P8722241760384591706_c_nat @ ( the_Pr7350365623720035563_c_nat @ Option ) ) )
              & ~ ( P2 @ ( F2 @ ( the_Pr7350365623720035563_c_nat @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_354_option_Odistinct_I1_J,axiom,
    ! [X2: produc4862256710654508797_c_nat] :
      ( none_P8487441334512977628_c_nat
     != ( some_P2720002978653898840_c_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_355_option_OdiscI,axiom,
    ! [Option: option2970150418924381261_c_nat,X2: produc4862256710654508797_c_nat] :
      ( ( Option
        = ( some_P2720002978653898840_c_nat @ X2 ) )
     => ( Option != none_P8487441334512977628_c_nat ) ) ).

% option.discI
thf(fact_356_option_Oexhaust,axiom,
    ! [Y4: option2970150418924381261_c_nat] :
      ( ( Y4 != none_P8487441334512977628_c_nat )
     => ~ ! [X23: produc4862256710654508797_c_nat] :
            ( Y4
           != ( some_P2720002978653898840_c_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_357_split__option__ex,axiom,
    ( ( ^ [P3: option2970150418924381261_c_nat > $o] :
        ? [X6: option2970150418924381261_c_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option2970150418924381261_c_nat > $o] :
          ( ( P4 @ none_P8487441334512977628_c_nat )
          | ? [X3: produc4862256710654508797_c_nat] : ( P4 @ ( some_P2720002978653898840_c_nat @ X3 ) ) ) ) ) ).

% split_option_ex
thf(fact_358_split__option__all,axiom,
    ( ( ^ [P3: option2970150418924381261_c_nat > $o] :
        ! [X6: option2970150418924381261_c_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option2970150418924381261_c_nat > $o] :
          ( ( P4 @ none_P8487441334512977628_c_nat )
          & ! [X3: produc4862256710654508797_c_nat] : ( P4 @ ( some_P2720002978653898840_c_nat @ X3 ) ) ) ) ) ).

% split_option_all
thf(fact_359_combine__options__cases,axiom,
    ! [X4: option2970150418924381261_c_nat,P2: option2970150418924381261_c_nat > option2970150418924381261_c_nat > $o,Y4: option2970150418924381261_c_nat] :
      ( ( ( X4 = none_P8487441334512977628_c_nat )
       => ( P2 @ X4 @ Y4 ) )
     => ( ( ( Y4 = none_P8487441334512977628_c_nat )
         => ( P2 @ X4 @ Y4 ) )
       => ( ! [A3: produc4862256710654508797_c_nat,B3: produc4862256710654508797_c_nat] :
              ( ( X4
                = ( some_P2720002978653898840_c_nat @ A3 ) )
             => ( ( Y4
                  = ( some_P2720002978653898840_c_nat @ B3 ) )
               => ( P2 @ X4 @ Y4 ) ) )
         => ( P2 @ X4 @ Y4 ) ) ) ) ).

% combine_options_cases
thf(fact_360_option_Osimps_I5_J,axiom,
    ! [F1: produc4862256710654508797_c_nat,F2: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,X2: produc4862256710654508797_c_nat] :
      ( ( case_o1602653449097480839_c_nat @ F1 @ F2 @ ( some_P2720002978653898840_c_nat @ X2 ) )
      = ( F2 @ X2 ) ) ).

% option.simps(5)
thf(fact_361_option_Osimps_I5_J,axiom,
    ! [F1: produc4862256710654508797_c_nat,F2: product_prod_c_nat > produc4862256710654508797_c_nat,X2: product_prod_c_nat] :
      ( ( case_o6498799718091186953_c_nat @ F1 @ F2 @ ( some_P8722241760384591706_c_nat @ X2 ) )
      = ( F2 @ X2 ) ) ).

% option.simps(5)
thf(fact_362_option_Oexpand,axiom,
    ! [Option: option2970150418924381261_c_nat,Option2: option2970150418924381261_c_nat] :
      ( ( ( Option = none_P8487441334512977628_c_nat )
        = ( Option2 = none_P8487441334512977628_c_nat ) )
     => ( ( ( Option != none_P8487441334512977628_c_nat )
         => ( ( Option2 != none_P8487441334512977628_c_nat )
           => ( ( the_Pr3263588667584626537_c_nat @ Option )
              = ( the_Pr3263588667584626537_c_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_363_Mapping__keys__dest,axiom,
    ! [X4: b,F: list_P7417839048565863355_c_nat] :
      ( ( member_b @ X4 @ ( mmap_k3080892797436903101_c_nat @ F ) )
     => ? [Y: produc4862256710654508797_c_nat] :
          ( ( mmap_l5026999719965937591_c_nat @ F @ X4 )
          = ( some_P2720002978653898840_c_nat @ Y ) ) ) ).

% Mapping_keys_dest
thf(fact_364_option_Ocase__eq__if,axiom,
    ( case_o1602653449097480839_c_nat
    = ( ^ [F12: produc4862256710654508797_c_nat,F22: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Option3: option2970150418924381261_c_nat] : ( if_Pro7338708846875558327_c_nat @ ( Option3 = none_P8487441334512977628_c_nat ) @ F12 @ ( F22 @ ( the_Pr3263588667584626537_c_nat @ Option3 ) ) ) ) ) ).

% option.case_eq_if
thf(fact_365_option_Ocase__eq__if,axiom,
    ( case_o6498799718091186953_c_nat
    = ( ^ [F12: produc4862256710654508797_c_nat,F22: product_prod_c_nat > produc4862256710654508797_c_nat,Option3: option7520157102916957007_c_nat] : ( if_Pro7338708846875558327_c_nat @ ( Option3 = none_P6573378090780909534_c_nat ) @ F12 @ ( F22 @ ( the_Pr7350365623720035563_c_nat @ Option3 ) ) ) ) ) ).

% option.case_eq_if
thf(fact_366_mmap__lookup__update_H,axiom,
    ! [K2: b,Z: b,V4: produc4862256710654508797_c_nat,Kvs: list_P7417839048565863355_c_nat] :
      ( ( ( K2 = Z )
       => ( ( mmap_l5026999719965937591_c_nat @ ( mmap_u2662500248198004552_c_nat @ K2 @ V4 @ Kvs ) @ Z )
          = ( some_P2720002978653898840_c_nat @ V4 ) ) )
      & ( ( K2 != Z )
       => ( ( mmap_l5026999719965937591_c_nat @ ( mmap_u2662500248198004552_c_nat @ K2 @ V4 @ Kvs ) @ Z )
          = ( mmap_l5026999719965937591_c_nat @ Kvs @ Z ) ) ) ) ).

% mmap_lookup_update'
thf(fact_367_mmap__lookup__update_H,axiom,
    ! [K2: b,Z: b,V4: c,Kvs: list_P903359562653991662od_b_c] :
      ( ( ( K2 = Z )
       => ( ( mmap_lookup_b_c @ ( mmap_update_b_c @ K2 @ V4 @ Kvs ) @ Z )
          = ( some_c @ V4 ) ) )
      & ( ( K2 != Z )
       => ( ( mmap_lookup_b_c @ ( mmap_update_b_c @ K2 @ V4 @ Kvs ) @ Z )
          = ( mmap_lookup_b_c @ Kvs @ Z ) ) ) ) ).

% mmap_lookup_update'
thf(fact_368_case__optionE,axiom,
    ! [P2: $o,Q2: produc4862256710654508797_c_nat > $o,X4: option2970150418924381261_c_nat] :
      ( ( case_o5420598178156702154_c_nat @ P2 @ Q2 @ X4 )
     => ( ( ( X4 = none_P8487441334512977628_c_nat )
         => ~ P2 )
       => ~ ! [Y: produc4862256710654508797_c_nat] :
              ( ( X4
                = ( some_P2720002978653898840_c_nat @ Y ) )
             => ~ ( Q2 @ Y ) ) ) ) ).

% case_optionE
thf(fact_369_image__eqI,axiom,
    ! [B: b,F: b > b,X4: b,A4: set_b] :
      ( ( B
        = ( F @ X4 ) )
     => ( ( member_b @ X4 @ A4 )
       => ( member_b @ B @ ( image_b_b @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_370_Compr__image__eq,axiom,
    ! [F: b > b,A4: set_b,P2: b > $o] :
      ( ( collect_b
        @ ^ [X3: b] :
            ( ( member_b @ X3 @ ( image_b_b @ F @ A4 ) )
            & ( P2 @ X3 ) ) )
      = ( image_b_b @ F
        @ ( collect_b
          @ ^ [X3: b] :
              ( ( member_b @ X3 @ A4 )
              & ( P2 @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_371_imageE,axiom,
    ! [B: b,F: b > b,A4: set_b] :
      ( ( member_b @ B @ ( image_b_b @ F @ A4 ) )
     => ~ ! [X: b] :
            ( ( B
              = ( F @ X ) )
           => ~ ( member_b @ X @ A4 ) ) ) ).

% imageE
thf(fact_372_None__notin__image__Some,axiom,
    ! [A4: set_Pr8806432033423503795_c_nat] :
      ~ ( member8863226656820029924_c_nat @ none_P8487441334512977628_c_nat @ ( image_1279218160517906513_c_nat @ some_P2720002978653898840_c_nat @ A4 ) ) ).

% None_notin_image_Some
thf(fact_373_rev__image__eqI,axiom,
    ! [X4: b,A4: set_b,B: b,F: b > b] :
      ( ( member_b @ X4 @ A4 )
     => ( ( B
          = ( F @ X4 ) )
       => ( member_b @ B @ ( image_b_b @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_374_imageI,axiom,
    ! [X4: b,A4: set_b,F: b > b] :
      ( ( member_b @ X4 @ A4 )
     => ( member_b @ ( F @ X4 ) @ ( image_b_b @ F @ A4 ) ) ) ).

% imageI
thf(fact_375_map__eq__map__tailrec,axiom,
    map_Pr6200325787298368846_nat_b = map_ta8214447623323088260_nat_b ).

% map_eq_map_tailrec
thf(fact_376_map__eq__map__tailrec,axiom,
    map_Pr8222292247188198875_b_c_b = map_ta599734118342162981_b_c_b ).

% map_eq_map_tailrec
thf(fact_377_Option_Othese__def,axiom,
    ( these_5226245170637726720_c_nat
    = ( ^ [A6: set_op6736743094888940163_c_nat] :
          ( image_1738307649304407377_c_nat @ the_Pr3263588667584626537_c_nat
          @ ( collec5647424909086326562_c_nat
            @ ^ [X3: option2970150418924381261_c_nat] :
                ( ( member8863226656820029924_c_nat @ X3 @ A6 )
                & ( X3 != none_P8487441334512977628_c_nat ) ) ) ) ) ) ).

% Option.these_def
thf(fact_378_Some__image__these__eq,axiom,
    ! [A4: set_op6736743094888940163_c_nat] :
      ( ( image_1279218160517906513_c_nat @ some_P2720002978653898840_c_nat @ ( these_5226245170637726720_c_nat @ A4 ) )
      = ( collec5647424909086326562_c_nat
        @ ^ [X3: option2970150418924381261_c_nat] :
            ( ( member8863226656820029924_c_nat @ X3 @ A4 )
            & ( X3 != none_P8487441334512977628_c_nat ) ) ) ) ).

% Some_image_these_eq
thf(fact_379_zip__map1,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,Ys: list_o9079336047566018143_c_nat] :
      ( ( zip_b_1039741285442616279_c_nat @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) @ Ys )
      = ( map_Pr9084924007856193940_c_nat
        @ ( produc5111080775214896549_c_nat
          @ ^ [X3: produc6689720525226317995_c_nat] : ( produc5716802255202478839_c_nat @ ( F @ X3 ) ) )
        @ ( zip_Pr2305811495203439539_c_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_380_zip__map1,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,Ys: list_o9079336047566018143_c_nat] :
      ( ( zip_b_1039741285442616279_c_nat @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) @ Ys )
      = ( map_Pr1659577239256917289_c_nat
        @ ( produc4989603208073378698_c_nat
          @ ^ [X3: product_prod_b_c] : ( produc5716802255202478839_c_nat @ ( F @ X3 ) ) )
        @ ( zip_Pr8888457682041370816_c_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_381_zip__map1,axiom,
    ! [F: b > b,Xs: list_b,Ys: list_o9079336047566018143_c_nat] :
      ( ( zip_b_1039741285442616279_c_nat @ ( map_b_b @ F @ Xs ) @ Ys )
      = ( map_Pr5967270692098401216_c_nat
        @ ( produc3722540595886809633_c_nat
          @ ^ [X3: b] : ( produc5716802255202478839_c_nat @ ( F @ X3 ) ) )
        @ ( zip_b_1039741285442616279_c_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_382_zip__map2,axiom,
    ! [Xs: list_b,F: option7520157102916957007_c_nat > option7520157102916957007_c_nat,Ys: list_o9079336047566018143_c_nat] :
      ( ( zip_b_1039741285442616279_c_nat @ Xs @ ( map_op8030519967798011748_c_nat @ F @ Ys ) )
      = ( map_Pr5967270692098401216_c_nat
        @ ( produc3722540595886809633_c_nat
          @ ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( produc5716802255202478839_c_nat @ X3 @ ( F @ Y3 ) ) )
        @ ( zip_b_1039741285442616279_c_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_383_zip__map__map,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,G: produc6689720525226317995_c_nat > b,Ys: list_P7417839048565863355_c_nat] :
      ( ( zip_b_b @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) @ ( map_Pr6200325787298368846_nat_b @ G @ Ys ) )
      = ( map_Pr7875562161635443252od_b_b
        @ ( produc3635197522347627491od_b_b
          @ ^ [X3: produc6689720525226317995_c_nat,Y3: produc6689720525226317995_c_nat] : ( product_Pair_b_b @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( zip_Pr2446451330630230287_c_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_384_zip__map__map,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,G: product_prod_b_c > b,Ys: list_P903359562653991662od_b_c] :
      ( ( zip_b_b @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) @ ( map_Pr8222292247188198875_b_c_b @ G @ Ys ) )
      = ( map_Pr734103314023365679od_b_b
        @ ( produc4235446546389705686od_b_b
          @ ^ [X3: produc6689720525226317995_c_nat,Y3: product_prod_b_c] : ( product_Pair_b_b @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( zip_Pr8181290086382856452od_b_c @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_385_zip__map__map,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,G: produc6689720525226317995_c_nat > b,Ys: list_P7417839048565863355_c_nat] :
      ( ( zip_b_b @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) @ ( map_Pr6200325787298368846_nat_b @ G @ Ys ) )
      = ( map_Pr5516162617174381407od_b_b
        @ ( produc223302539832192702od_b_b
          @ ^ [X3: product_prod_b_c,Y3: produc6689720525226317995_c_nat] : ( product_Pair_b_b @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( zip_Pr8387323144706649116_c_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_386_zip__map__map,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,G: product_prod_b_c > b,Ys: list_P903359562653991662od_b_c] :
      ( ( zip_b_b @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) @ ( map_Pr8222292247188198875_b_c_b @ G @ Ys ) )
      = ( map_Pr4341431725133323588od_b_b
        @ ( produc1457153794959892283od_b_b
          @ ^ [X3: product_prod_b_c,Y3: product_prod_b_c] : ( product_Pair_b_b @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( zip_Pr9032920399426446775od_b_c @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_387_zip__map__map,axiom,
    ! [F: b > b,Xs: list_b,G: option7520157102916957007_c_nat > option7520157102916957007_c_nat,Ys: list_o9079336047566018143_c_nat] :
      ( ( zip_b_1039741285442616279_c_nat @ ( map_b_b @ F @ Xs ) @ ( map_op8030519967798011748_c_nat @ G @ Ys ) )
      = ( map_Pr5967270692098401216_c_nat
        @ ( produc3722540595886809633_c_nat
          @ ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( produc5716802255202478839_c_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( zip_b_1039741285442616279_c_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_388_zip__map__map,axiom,
    ! [F: c > b,Xs: list_c,G: nat > option7520157102916957007_c_nat,Ys: list_nat] :
      ( ( zip_b_1039741285442616279_c_nat @ ( map_c_b @ F @ Xs ) @ ( map_na158159703617048037_c_nat @ G @ Ys ) )
      = ( map_Pr6194916426178313410_c_nat
        @ ( produc3293230613241633187_c_nat
          @ ^ [X3: c,Y3: nat] : ( produc5716802255202478839_c_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( zip_c_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_389_image__paired__Times,axiom,
    ! [F: b > b,G: option7520157102916957007_c_nat > option7520157102916957007_c_nat,A4: set_b,B5: set_op6398572775360143365_c_nat] :
      ( ( image_7713823756821623041_c_nat
        @ ( produc3722540595886809633_c_nat
          @ ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( produc5716802255202478839_c_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( produc1892124643397453190_c_nat @ A4
          @ ^ [Uu: b] : B5 ) )
      = ( produc1892124643397453190_c_nat @ ( image_b_b @ F @ A4 )
        @ ^ [Uu: b] : ( image_6555862624884852389_c_nat @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_390_image__paired__Times,axiom,
    ! [F: c > b,G: nat > option7520157102916957007_c_nat,A4: set_c,B5: set_nat] :
      ( ( image_1176608181733484675_c_nat
        @ ( produc3293230613241633187_c_nat
          @ ^ [X3: c,Y3: nat] : ( produc5716802255202478839_c_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( product_Sigma_c_nat @ A4
          @ ^ [Uu: c] : B5 ) )
      = ( produc1892124643397453190_c_nat @ ( image_c_b @ F @ A4 )
        @ ^ [Uu: b] : ( image_6113942820217916326_c_nat @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_391_map__zip__map2,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_b,G: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Ys: list_P8155366108056616461_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ F @ ( zip_b_1292320801234550661_c_nat @ Xs @ ( map_Pr5967270692098401216_c_nat @ G @ Ys ) ) )
      = ( map_Pr6200325787298368846_nat_b
        @ ( produc1702926192408969901_nat_b
          @ ^ [X3: b,Y3: produc4862256710654508797_c_nat] : ( F @ ( produc8868828497325608613_c_nat @ X3 @ ( G @ Y3 ) ) ) )
        @ ( zip_b_1292320801234550661_c_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_392_map__zip__map2,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_b,G: c > produc4862256710654508797_c_nat,Ys: list_c] :
      ( ( map_Pr6200325787298368846_nat_b @ F @ ( zip_b_1292320801234550661_c_nat @ Xs @ ( map_c_816562529315836601_c_nat @ G @ Ys ) ) )
      = ( map_Pr8222292247188198875_b_c_b
        @ ( produc8837194378803474130_b_c_b
          @ ^ [X3: b,Y3: c] : ( F @ ( produc8868828497325608613_c_nat @ X3 @ ( G @ Y3 ) ) ) )
        @ ( zip_b_c @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_393_map__zip__map2,axiom,
    ! [F: product_prod_b_c > b,Xs: list_b,G: produc4862256710654508797_c_nat > c,Ys: list_P8155366108056616461_c_nat] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( zip_b_c @ Xs @ ( map_Pr418519014202471037_nat_c @ G @ Ys ) ) )
      = ( map_Pr6200325787298368846_nat_b
        @ ( produc1702926192408969901_nat_b
          @ ^ [X3: b,Y3: produc4862256710654508797_c_nat] : ( F @ ( product_Pair_b_c @ X3 @ ( G @ Y3 ) ) ) )
        @ ( zip_b_1292320801234550661_c_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_394_map__zip__map2,axiom,
    ! [F: product_prod_b_c > b,Xs: list_b,G: c > c,Ys: list_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( zip_b_c @ Xs @ ( map_c_c @ G @ Ys ) ) )
      = ( map_Pr8222292247188198875_b_c_b
        @ ( produc8837194378803474130_b_c_b
          @ ^ [X3: b,Y3: c] : ( F @ ( product_Pair_b_c @ X3 @ ( G @ Y3 ) ) ) )
        @ ( zip_b_c @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_395_map__zip__map2,axiom,
    ! [F: produc4862256710654508797_c_nat > b,Xs: list_b,G: produc4862256710654508797_c_nat > option7520157102916957007_c_nat,Ys: list_P8155366108056616461_c_nat] :
      ( ( map_Pr418519014202471036_nat_b @ F @ ( zip_b_1039741285442616279_c_nat @ Xs @ ( map_Pr8931211203828011794_c_nat @ G @ Ys ) ) )
      = ( map_Pr6200325787298368846_nat_b
        @ ( produc1702926192408969901_nat_b
          @ ^ [X3: b,Y3: produc4862256710654508797_c_nat] : ( F @ ( produc5716802255202478839_c_nat @ X3 @ ( G @ Y3 ) ) ) )
        @ ( zip_b_1292320801234550661_c_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_396_map__zip__map2,axiom,
    ! [F: produc4862256710654508797_c_nat > b,Xs: list_b,G: c > option7520157102916957007_c_nat,Ys: list_c] :
      ( ( map_Pr418519014202471036_nat_b @ F @ ( zip_b_1039741285442616279_c_nat @ Xs @ ( map_c_5865653670955426571_c_nat @ G @ Ys ) ) )
      = ( map_Pr8222292247188198875_b_c_b
        @ ( produc8837194378803474130_b_c_b
          @ ^ [X3: b,Y3: c] : ( F @ ( produc5716802255202478839_c_nat @ X3 @ ( G @ Y3 ) ) ) )
        @ ( zip_b_c @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_397_map__zip__map2,axiom,
    ! [F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Xs: list_b,G: option7520157102916957007_c_nat > option7520157102916957007_c_nat,Ys: list_o9079336047566018143_c_nat] :
      ( ( map_Pr5967270692098401216_c_nat @ F @ ( zip_b_1039741285442616279_c_nat @ Xs @ ( map_op8030519967798011748_c_nat @ G @ Ys ) ) )
      = ( map_Pr5967270692098401216_c_nat
        @ ( produc3722540595886809633_c_nat
          @ ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( F @ ( produc5716802255202478839_c_nat @ X3 @ ( G @ Y3 ) ) ) )
        @ ( zip_b_1039741285442616279_c_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_398_map__zip__map2,axiom,
    ! [F: product_prod_c_nat > produc4862256710654508797_c_nat,Xs: list_c,G: nat > nat,Ys: list_nat] :
      ( ( map_Pr6194916426178313410_c_nat @ F @ ( zip_c_nat @ Xs @ ( map_nat_nat @ G @ Ys ) ) )
      = ( map_Pr6194916426178313410_c_nat
        @ ( produc3293230613241633187_c_nat
          @ ^ [X3: c,Y3: nat] : ( F @ ( product_Pair_c_nat @ X3 @ ( G @ Y3 ) ) ) )
        @ ( zip_c_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_399_map__zip__map,axiom,
    ! [F: produc6689720525226317995_c_nat > b,G: b > b,Xs: list_b,Ys: list_P8155366108056616461_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ F @ ( zip_b_1292320801234550661_c_nat @ ( map_b_b @ G @ Xs ) @ Ys ) )
      = ( map_Pr6200325787298368846_nat_b
        @ ( produc1702926192408969901_nat_b
          @ ^ [X3: b,Y3: produc4862256710654508797_c_nat] : ( F @ ( produc8868828497325608613_c_nat @ ( G @ X3 ) @ Y3 ) ) )
        @ ( zip_b_1292320801234550661_c_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_400_map__zip__map,axiom,
    ! [F: produc6689720525226317995_c_nat > b,G: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,Ys: list_P8155366108056616461_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ F @ ( zip_b_1292320801234550661_c_nat @ ( map_Pr6200325787298368846_nat_b @ G @ Xs ) @ Ys ) )
      = ( map_Pr8419796046120434298_nat_b
        @ ( produc1828352386651926057_nat_b
          @ ^ [X3: produc6689720525226317995_c_nat,Y3: produc4862256710654508797_c_nat] : ( F @ ( produc8868828497325608613_c_nat @ ( G @ X3 ) @ Y3 ) ) )
        @ ( zip_Pr3260927158048934753_c_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_401_map__zip__map,axiom,
    ! [F: produc6689720525226317995_c_nat > b,G: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,Ys: list_P8155366108056616461_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ F @ ( zip_b_1292320801234550661_c_nat @ ( map_Pr8222292247188198875_b_c_b @ G @ Xs ) @ Ys ) )
      = ( map_Pr3906394697118060965_nat_b
        @ ( produc7091333456633343748_nat_b
          @ ^ [X3: product_prod_b_c,Y3: produc4862256710654508797_c_nat] : ( F @ ( produc8868828497325608613_c_nat @ ( G @ X3 ) @ Y3 ) ) )
        @ ( zip_Pr1917056362909182830_c_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_402_map__zip__map,axiom,
    ! [F: product_prod_b_c > b,G: b > b,Xs: list_b,Ys: list_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( zip_b_c @ ( map_b_b @ G @ Xs ) @ Ys ) )
      = ( map_Pr8222292247188198875_b_c_b
        @ ( produc8837194378803474130_b_c_b
          @ ^ [X3: b,Y3: c] : ( F @ ( product_Pair_b_c @ ( G @ X3 ) @ Y3 ) ) )
        @ ( zip_b_c @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_403_map__zip__map,axiom,
    ! [F: product_prod_b_c > b,G: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,Ys: list_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( zip_b_c @ ( map_Pr6200325787298368846_nat_b @ G @ Xs ) @ Ys ) )
      = ( map_Pr4195343743593861551at_c_b
        @ ( produc1549832927761746006at_c_b
          @ ^ [X3: produc6689720525226317995_c_nat,Y3: c] : ( F @ ( product_Pair_b_c @ ( G @ X3 ) @ Y3 ) ) )
        @ ( zip_Pr6015829939078101404_nat_c @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_404_map__zip__map,axiom,
    ! [F: product_prod_b_c > b,G: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,Ys: list_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( zip_b_c @ ( map_Pr8222292247188198875_b_c_b @ G @ Xs ) @ Ys ) )
      = ( map_Pr5401617559127108804_c_c_b
        @ ( produc5378556977916894651_c_c_b
          @ ^ [X3: product_prod_b_c,Y3: c] : ( F @ ( product_Pair_b_c @ ( G @ X3 ) @ Y3 ) ) )
        @ ( zip_Pr4210574531626869071_b_c_c @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_405_map__zip__map,axiom,
    ! [F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,G: b > b,Xs: list_b,Ys: list_o9079336047566018143_c_nat] :
      ( ( map_Pr5967270692098401216_c_nat @ F @ ( zip_b_1039741285442616279_c_nat @ ( map_b_b @ G @ Xs ) @ Ys ) )
      = ( map_Pr5967270692098401216_c_nat
        @ ( produc3722540595886809633_c_nat
          @ ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( F @ ( produc5716802255202478839_c_nat @ ( G @ X3 ) @ Y3 ) ) )
        @ ( zip_b_1039741285442616279_c_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_406_map__zip__map,axiom,
    ! [F: product_prod_c_nat > produc4862256710654508797_c_nat,G: c > c,Xs: list_c,Ys: list_nat] :
      ( ( map_Pr6194916426178313410_c_nat @ F @ ( zip_c_nat @ ( map_c_c @ G @ Xs ) @ Ys ) )
      = ( map_Pr6194916426178313410_c_nat
        @ ( produc3293230613241633187_c_nat
          @ ^ [X3: c,Y3: nat] : ( F @ ( product_Pair_c_nat @ ( G @ X3 ) @ Y3 ) ) )
        @ ( zip_c_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_407_mem__Sigma__iff,axiom,
    ! [A: b,B: b,A4: set_b,B5: b > set_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( product_Sigma_b_b @ A4 @ B5 ) )
      = ( ( member_b @ A @ A4 )
        & ( member_b @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_408_mem__Sigma__iff,axiom,
    ! [A: b,B: option7520157102916957007_c_nat,A4: set_b,B5: b > set_op6398572775360143365_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A @ B ) @ ( produc1892124643397453190_c_nat @ A4 @ B5 ) )
      = ( ( member_b @ A @ A4 )
        & ( member3208607818295432806_c_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_409_mem__Sigma__iff,axiom,
    ! [A: c,B: nat,A4: set_c,B5: c > set_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A @ B ) @ ( product_Sigma_c_nat @ A4 @ B5 ) )
      = ( ( member_c @ A @ A4 )
        & ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_410_SigmaI,axiom,
    ! [A: b,A4: set_b,B: b,B5: b > set_b] :
      ( ( member_b @ A @ A4 )
     => ( ( member_b @ B @ ( B5 @ A ) )
       => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( product_Sigma_b_b @ A4 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_411_SigmaI,axiom,
    ! [A: b,A4: set_b,B: option7520157102916957007_c_nat,B5: b > set_op6398572775360143365_c_nat] :
      ( ( member_b @ A @ A4 )
     => ( ( member3208607818295432806_c_nat @ B @ ( B5 @ A ) )
       => ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A @ B ) @ ( produc1892124643397453190_c_nat @ A4 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_412_SigmaI,axiom,
    ! [A: c,A4: set_c,B: nat,B5: c > set_nat] :
      ( ( member_c @ A @ A4 )
     => ( ( member_nat @ B @ ( B5 @ A ) )
       => ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A @ B ) @ ( product_Sigma_c_nat @ A4 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_413_SigmaE2,axiom,
    ! [A: b,B: b,A4: set_b,B5: b > set_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( product_Sigma_b_b @ A4 @ B5 ) )
     => ~ ( ( member_b @ A @ A4 )
         => ~ ( member_b @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_414_SigmaE2,axiom,
    ! [A: b,B: option7520157102916957007_c_nat,A4: set_b,B5: b > set_op6398572775360143365_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A @ B ) @ ( produc1892124643397453190_c_nat @ A4 @ B5 ) )
     => ~ ( ( member_b @ A @ A4 )
         => ~ ( member3208607818295432806_c_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_415_SigmaE2,axiom,
    ! [A: c,B: nat,A4: set_c,B5: c > set_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A @ B ) @ ( product_Sigma_c_nat @ A4 @ B5 ) )
     => ~ ( ( member_c @ A @ A4 )
         => ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_416_SigmaD2,axiom,
    ! [A: b,B: option7520157102916957007_c_nat,A4: set_b,B5: b > set_op6398572775360143365_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A @ B ) @ ( produc1892124643397453190_c_nat @ A4 @ B5 ) )
     => ( member3208607818295432806_c_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_417_SigmaD2,axiom,
    ! [A: c,B: nat,A4: set_c,B5: c > set_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A @ B ) @ ( product_Sigma_c_nat @ A4 @ B5 ) )
     => ( member_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_418_SigmaD1,axiom,
    ! [A: b,B: option7520157102916957007_c_nat,A4: set_b,B5: b > set_op6398572775360143365_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A @ B ) @ ( produc1892124643397453190_c_nat @ A4 @ B5 ) )
     => ( member_b @ A @ A4 ) ) ).

% SigmaD1
thf(fact_419_SigmaD1,axiom,
    ! [A: c,B: nat,A4: set_c,B5: c > set_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A @ B ) @ ( product_Sigma_c_nat @ A4 @ B5 ) )
     => ( member_c @ A @ A4 ) ) ).

% SigmaD1
thf(fact_420_SigmaE,axiom,
    ! [C: product_prod_b_b,A4: set_b,B5: b > set_b] :
      ( ( member7862447936710763792od_b_b @ C @ ( product_Sigma_b_b @ A4 @ B5 ) )
     => ~ ! [X: b] :
            ( ( member_b @ X @ A4 )
           => ! [Y: b] :
                ( ( member_b @ Y @ ( B5 @ X ) )
               => ( C
                 != ( product_Pair_b_b @ X @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_421_SigmaE,axiom,
    ! [C: produc4862256710654508797_c_nat,A4: set_b,B5: b > set_op6398572775360143365_c_nat] :
      ( ( member7562873241046315796_c_nat @ C @ ( produc1892124643397453190_c_nat @ A4 @ B5 ) )
     => ~ ! [X: b] :
            ( ( member_b @ X @ A4 )
           => ! [Y: option7520157102916957007_c_nat] :
                ( ( member3208607818295432806_c_nat @ Y @ ( B5 @ X ) )
               => ( C
                 != ( produc5716802255202478839_c_nat @ X @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_422_SigmaE,axiom,
    ! [C: product_prod_c_nat,A4: set_c,B5: c > set_nat] :
      ( ( member8195077246299207702_c_nat @ C @ ( product_Sigma_c_nat @ A4 @ B5 ) )
     => ~ ! [X: c] :
            ( ( member_c @ X @ A4 )
           => ! [Y: nat] :
                ( ( member_nat @ Y @ ( B5 @ X ) )
               => ( C
                 != ( product_Pair_c_nat @ X @ Y ) ) ) ) ) ).

% SigmaE
thf(fact_423_in__these__eq,axiom,
    ! [X4: b,A4: set_option_b] :
      ( ( member_b @ X4 @ ( these_b @ A4 ) )
      = ( member_option_b @ ( some_b @ X4 ) @ A4 ) ) ).

% in_these_eq
thf(fact_424_map2__map__map,axiom,
    ! [H: b > b > b,F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,G: produc6689720525226317995_c_nat > b] :
      ( ( map_Pr8857941247793185564_b_b_b @ ( produc2401277923921374355_b_b_b @ H ) @ ( zip_b_b @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) @ ( map_Pr6200325787298368846_nat_b @ G @ Xs ) ) )
      = ( map_Pr6200325787298368846_nat_b
        @ ^ [X3: produc6689720525226317995_c_nat] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_425_map2__map__map,axiom,
    ! [H: b > b > b,F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,G: product_prod_b_c > b] :
      ( ( map_Pr8857941247793185564_b_b_b @ ( produc2401277923921374355_b_b_b @ H ) @ ( zip_b_b @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) @ ( map_Pr8222292247188198875_b_c_b @ G @ Xs ) ) )
      = ( map_Pr8222292247188198875_b_c_b
        @ ^ [X3: product_prod_b_c] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_426_map2__map__map,axiom,
    ! [H: b > produc4862256710654508797_c_nat > b,F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,G: produc6689720525226317995_c_nat > produc4862256710654508797_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ ( produc1702926192408969901_nat_b @ H ) @ ( zip_b_1292320801234550661_c_nat @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) @ ( map_Pr4595408008933130094_c_nat @ G @ Xs ) ) )
      = ( map_Pr6200325787298368846_nat_b
        @ ^ [X3: produc6689720525226317995_c_nat] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_427_map2__map__map,axiom,
    ! [H: b > produc4862256710654508797_c_nat > b,F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,G: product_prod_b_c > produc4862256710654508797_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ ( produc1702926192408969901_nat_b @ H ) @ ( zip_b_1292320801234550661_c_nat @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) @ ( map_Pr2101552211129450273_c_nat @ G @ Xs ) ) )
      = ( map_Pr8222292247188198875_b_c_b
        @ ^ [X3: product_prod_b_c] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_428_map2__map__map,axiom,
    ! [H: b > c > b,F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,G: produc6689720525226317995_c_nat > c] :
      ( ( map_Pr8222292247188198875_b_c_b @ ( produc8837194378803474130_b_c_b @ H ) @ ( zip_b_c @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) @ ( map_Pr6200325787298368847_nat_c @ G @ Xs ) ) )
      = ( map_Pr6200325787298368846_nat_b
        @ ^ [X3: produc6689720525226317995_c_nat] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_429_map2__map__map,axiom,
    ! [H: b > c > b,F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,G: product_prod_b_c > c] :
      ( ( map_Pr8222292247188198875_b_c_b @ ( produc8837194378803474130_b_c_b @ H ) @ ( zip_b_c @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) @ ( map_Pr8222292247188198876_b_c_c @ G @ Xs ) ) )
      = ( map_Pr8222292247188198875_b_c_b
        @ ^ [X3: product_prod_b_c] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_430_map2__map__map,axiom,
    ! [H: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,G: produc6689720525226317995_c_nat > option7520157102916957007_c_nat] :
      ( ( map_Pr5967270692098401216_c_nat @ ( produc3722540595886809633_c_nat @ H ) @ ( zip_b_1039741285442616279_c_nat @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) @ ( map_Pr1008512146171366592_c_nat @ G @ Xs ) ) )
      = ( map_Pr4595408008933130094_c_nat
        @ ^ [X3: produc6689720525226317995_c_nat] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_431_map2__map__map,axiom,
    ! [H: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,G: product_prod_b_c > option7520157102916957007_c_nat] :
      ( ( map_Pr5967270692098401216_c_nat @ ( produc3722540595886809633_c_nat @ H ) @ ( zip_b_1039741285442616279_c_nat @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) @ ( map_Pr4815041605510051699_c_nat @ G @ Xs ) ) )
      = ( map_Pr2101552211129450273_c_nat
        @ ^ [X3: product_prod_b_c] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_432_zip__commute,axiom,
    ( zip_b_1039741285442616279_c_nat
    = ( ^ [Xs2: list_b,Ys2: list_o9079336047566018143_c_nat] :
          ( map_Pr1687081633635275944_c_nat
          @ ( produc2411843245718984257_c_nat
            @ ^ [X3: option7520157102916957007_c_nat,Y3: b] : ( produc5716802255202478839_c_nat @ Y3 @ X3 ) )
          @ ( zip_op5169119592995119095_nat_b @ Ys2 @ Xs2 ) ) ) ) ).

% zip_commute
thf(fact_433_zip__commute,axiom,
    ( zip_c_nat
    = ( ^ [Xs2: list_c,Ys2: list_nat] :
          ( map_Pr4542497410526915306_c_nat
          @ ( produc8194310905338674051_c_nat
            @ ^ [X3: nat,Y3: c] : ( product_Pair_c_nat @ Y3 @ X3 ) )
          @ ( zip_nat_c @ Ys2 @ Xs2 ) ) ) ) ).

% zip_commute
thf(fact_434_swap__product,axiom,
    ! [A4: set_op6398572775360143365_c_nat,B5: set_b] :
      ( ( image_3433634698358497769_c_nat
        @ ( produc2411843245718984257_c_nat
          @ ^ [I: option7520157102916957007_c_nat,J: b] : ( produc5716802255202478839_c_nat @ J @ I ) )
        @ ( produc6021502950949956006_nat_b @ A4
          @ ^ [Uu: option7520157102916957007_c_nat] : B5 ) )
      = ( produc1892124643397453190_c_nat @ B5
        @ ^ [Uu: b] : A4 ) ) ).

% swap_product
thf(fact_435_swap__product,axiom,
    ! [A4: set_nat,B5: set_c] :
      ( ( image_6275841512968140075_c_nat
        @ ( produc8194310905338674051_c_nat
          @ ^ [I: nat,J: c] : ( product_Pair_c_nat @ J @ I ) )
        @ ( product_Sigma_nat_c @ A4
          @ ^ [Uu: nat] : B5 ) )
      = ( product_Sigma_c_nat @ B5
        @ ^ [Uu: c] : A4 ) ) ).

% swap_product
thf(fact_436_notin__range__Some,axiom,
    ! [X4: option2970150418924381261_c_nat] :
      ( ( ~ ( member8863226656820029924_c_nat @ X4 @ ( image_1279218160517906513_c_nat @ some_P2720002978653898840_c_nat @ top_to7683274595066931299_c_nat ) ) )
      = ( X4 = none_P8487441334512977628_c_nat ) ) ).

% notin_range_Some
thf(fact_437_map__ran__def,axiom,
    ( map_ra1379142644618337576_c_nat
    = ( ^ [F3: b > option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
          ( map_Pr5967270692098401216_c_nat
          @ ( produc3722540595886809633_c_nat
            @ ^ [K: b,V: option7520157102916957007_c_nat] : ( produc5716802255202478839_c_nat @ K @ ( F3 @ K @ V ) ) ) ) ) ) ).

% map_ran_def
thf(fact_438_map__prod__simp,axiom,
    ! [F: b > b,G: option7520157102916957007_c_nat > option7520157102916957007_c_nat,A: b,B: option7520157102916957007_c_nat] :
      ( ( produc3763450287754571990_c_nat @ F @ G @ ( produc5716802255202478839_c_nat @ A @ B ) )
      = ( produc5716802255202478839_c_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_439_map__prod__simp,axiom,
    ! [F: b > c,G: option7520157102916957007_c_nat > nat,A: b,B: option7520157102916957007_c_nat] :
      ( ( produc1238995859230195800at_nat @ F @ G @ ( produc5716802255202478839_c_nat @ A @ B ) )
      = ( product_Pair_c_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_440_map__prod__simp,axiom,
    ! [F: c > b,G: nat > option7520157102916957007_c_nat,A: c,B: nat] :
      ( ( produc8240166225308521750_c_nat @ F @ G @ ( product_Pair_c_nat @ A @ B ) )
      = ( produc5716802255202478839_c_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_441_map__prod__simp,axiom,
    ! [F: c > c,G: nat > nat,A: c,B: nat] :
      ( ( produc3634155571782648600at_nat @ F @ G @ ( product_Pair_c_nat @ A @ B ) )
      = ( product_Pair_c_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_442_None__eq__map__option__iff,axiom,
    ! [F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,X4: option2970150418924381261_c_nat] :
      ( ( none_P8487441334512977628_c_nat
        = ( map_op4586031498963288711_c_nat @ F @ X4 ) )
      = ( X4 = none_P8487441334512977628_c_nat ) ) ).

% None_eq_map_option_iff
thf(fact_443_map__option__is__None,axiom,
    ! [F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Opt: option2970150418924381261_c_nat] :
      ( ( ( map_op4586031498963288711_c_nat @ F @ Opt )
        = none_P8487441334512977628_c_nat )
      = ( Opt = none_P8487441334512977628_c_nat ) ) ).

% map_option_is_None
thf(fact_444_option_Omap__disc__iff,axiom,
    ! [F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,A: option2970150418924381261_c_nat] :
      ( ( ( map_op4586031498963288711_c_nat @ F @ A )
        = none_P8487441334512977628_c_nat )
      = ( A = none_P8487441334512977628_c_nat ) ) ).

% option.map_disc_iff
thf(fact_445_map__prod__imageI,axiom,
    ! [A: b,B: option7520157102916957007_c_nat,R: set_Pr8806432033423503795_c_nat,F: b > b,G: option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A @ B ) @ R )
     => ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_7713823756821623041_c_nat @ ( produc3763450287754571990_c_nat @ F @ G ) @ R ) ) ) ).

% map_prod_imageI
thf(fact_446_map__prod__imageI,axiom,
    ! [A: b,B: option7520157102916957007_c_nat,R: set_Pr8806432033423503795_c_nat,F: b > c,G: option7520157102916957007_c_nat > nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A @ B ) @ R )
     => ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_4213855687810486915_c_nat @ ( produc1238995859230195800at_nat @ F @ G ) @ R ) ) ) ).

% map_prod_imageI
thf(fact_447_map__prod__imageI,axiom,
    ! [A: c,B: nat,R: set_Pr6903500605879609269_c_nat,F: c > b,G: nat > option7520157102916957007_c_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A @ B ) @ R )
     => ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_1176608181733484675_c_nat @ ( produc8240166225308521750_c_nat @ F @ G ) @ R ) ) ) ).

% map_prod_imageI
thf(fact_448_map__prod__imageI,axiom,
    ! [A: c,B: nat,R: set_Pr6903500605879609269_c_nat,F: c > c,G: nat > nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A @ B ) @ R )
     => ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_1019079544199661317_c_nat @ ( produc3634155571782648600at_nat @ F @ G ) @ R ) ) ) ).

% map_prod_imageI
thf(fact_449_option_Osimps_I8_J,axiom,
    ! [F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( map_op4586031498963288711_c_nat @ F @ none_P8487441334512977628_c_nat )
      = none_P8487441334512977628_c_nat ) ).

% option.simps(8)
thf(fact_450_case__prod__map__prod,axiom,
    ! [H: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,F: b > b,G: option7520157102916957007_c_nat > option7520157102916957007_c_nat,X4: produc4862256710654508797_c_nat] :
      ( ( produc3722540595886809633_c_nat @ H @ ( produc3763450287754571990_c_nat @ F @ G @ X4 ) )
      = ( produc3722540595886809633_c_nat
        @ ^ [L: b,R3: option7520157102916957007_c_nat] : ( H @ ( F @ L ) @ ( G @ R3 ) )
        @ X4 ) ) ).

% case_prod_map_prod
thf(fact_451_case__prod__map__prod,axiom,
    ! [H: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,F: c > b,G: nat > option7520157102916957007_c_nat,X4: product_prod_c_nat] :
      ( ( produc3722540595886809633_c_nat @ H @ ( produc8240166225308521750_c_nat @ F @ G @ X4 ) )
      = ( produc3293230613241633187_c_nat
        @ ^ [L: c,R3: nat] : ( H @ ( F @ L ) @ ( G @ R3 ) )
        @ X4 ) ) ).

% case_prod_map_prod
thf(fact_452_case__prod__map__prod,axiom,
    ! [H: c > nat > produc4862256710654508797_c_nat,F: b > c,G: option7520157102916957007_c_nat > nat,X4: produc4862256710654508797_c_nat] :
      ( ( produc3293230613241633187_c_nat @ H @ ( produc1238995859230195800at_nat @ F @ G @ X4 ) )
      = ( produc3722540595886809633_c_nat
        @ ^ [L: b,R3: option7520157102916957007_c_nat] : ( H @ ( F @ L ) @ ( G @ R3 ) )
        @ X4 ) ) ).

% case_prod_map_prod
thf(fact_453_case__prod__map__prod,axiom,
    ! [H: c > nat > produc4862256710654508797_c_nat,F: c > c,G: nat > nat,X4: product_prod_c_nat] :
      ( ( produc3293230613241633187_c_nat @ H @ ( produc3634155571782648600at_nat @ F @ G @ X4 ) )
      = ( produc3293230613241633187_c_nat
        @ ^ [L: c,R3: nat] : ( H @ ( F @ L ) @ ( G @ R3 ) )
        @ X4 ) ) ).

% case_prod_map_prod
thf(fact_454_option_Omap__sel,axiom,
    ! [A: option2970150418924381261_c_nat,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( A != none_P8487441334512977628_c_nat )
     => ( ( the_Pr3263588667584626537_c_nat @ ( map_op4586031498963288711_c_nat @ F @ A ) )
        = ( F @ ( the_Pr3263588667584626537_c_nat @ A ) ) ) ) ).

% option.map_sel
thf(fact_455_prod__fun__imageE,axiom,
    ! [C: produc4862256710654508797_c_nat,F: b > b,G: option7520157102916957007_c_nat > option7520157102916957007_c_nat,R: set_Pr8806432033423503795_c_nat] :
      ( ( member7562873241046315796_c_nat @ C @ ( image_7713823756821623041_c_nat @ ( produc3763450287754571990_c_nat @ F @ G ) @ R ) )
     => ~ ! [X: b,Y: option7520157102916957007_c_nat] :
            ( ( C
              = ( produc5716802255202478839_c_nat @ ( F @ X ) @ ( G @ Y ) ) )
           => ~ ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X @ Y ) @ R ) ) ) ).

% prod_fun_imageE
thf(fact_456_prod__fun__imageE,axiom,
    ! [C: produc4862256710654508797_c_nat,F: c > b,G: nat > option7520157102916957007_c_nat,R: set_Pr6903500605879609269_c_nat] :
      ( ( member7562873241046315796_c_nat @ C @ ( image_1176608181733484675_c_nat @ ( produc8240166225308521750_c_nat @ F @ G ) @ R ) )
     => ~ ! [X: c,Y: nat] :
            ( ( C
              = ( produc5716802255202478839_c_nat @ ( F @ X ) @ ( G @ Y ) ) )
           => ~ ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X @ Y ) @ R ) ) ) ).

% prod_fun_imageE
thf(fact_457_prod__fun__imageE,axiom,
    ! [C: product_prod_c_nat,F: b > c,G: option7520157102916957007_c_nat > nat,R: set_Pr8806432033423503795_c_nat] :
      ( ( member8195077246299207702_c_nat @ C @ ( image_4213855687810486915_c_nat @ ( produc1238995859230195800at_nat @ F @ G ) @ R ) )
     => ~ ! [X: b,Y: option7520157102916957007_c_nat] :
            ( ( C
              = ( product_Pair_c_nat @ ( F @ X ) @ ( G @ Y ) ) )
           => ~ ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X @ Y ) @ R ) ) ) ).

% prod_fun_imageE
thf(fact_458_prod__fun__imageE,axiom,
    ! [C: product_prod_c_nat,F: c > c,G: nat > nat,R: set_Pr6903500605879609269_c_nat] :
      ( ( member8195077246299207702_c_nat @ C @ ( image_1019079544199661317_c_nat @ ( produc3634155571782648600at_nat @ F @ G ) @ R ) )
     => ~ ! [X: c,Y: nat] :
            ( ( C
              = ( product_Pair_c_nat @ ( F @ X ) @ ( G @ Y ) ) )
           => ~ ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X @ Y ) @ R ) ) ) ).

% prod_fun_imageE
thf(fact_459_map__prod__def,axiom,
    ( produc3763450287754571990_c_nat
    = ( ^ [F3: b > b,G2: option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
          ( produc3722540595886809633_c_nat
          @ ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( produc5716802255202478839_c_nat @ ( F3 @ X3 ) @ ( G2 @ Y3 ) ) ) ) ) ).

% map_prod_def
thf(fact_460_map__prod__def,axiom,
    ( produc8240166225308521750_c_nat
    = ( ^ [F3: c > b,G2: nat > option7520157102916957007_c_nat] :
          ( produc3293230613241633187_c_nat
          @ ^ [X3: c,Y3: nat] : ( produc5716802255202478839_c_nat @ ( F3 @ X3 ) @ ( G2 @ Y3 ) ) ) ) ) ).

% map_prod_def
thf(fact_461_UNIV__option__conv,axiom,
    ( top_to5886104162681696563_c_nat
    = ( insert6554936906190869117_c_nat @ none_P8487441334512977628_c_nat @ ( image_1279218160517906513_c_nat @ some_P2720002978653898840_c_nat @ top_to7683274595066931299_c_nat ) ) ) ).

% UNIV_option_conv
thf(fact_462_these__insert__None,axiom,
    ! [A4: set_op6736743094888940163_c_nat] :
      ( ( these_5226245170637726720_c_nat @ ( insert6554936906190869117_c_nat @ none_P8487441334512977628_c_nat @ A4 ) )
      = ( these_5226245170637726720_c_nat @ A4 ) ) ).

% these_insert_None
thf(fact_463_insert__compr,axiom,
    ( insert_b
    = ( ^ [A5: b,B6: set_b] :
          ( collect_b
          @ ^ [X3: b] :
              ( ( X3 = A5 )
              | ( member_b @ X3 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_464_insert__Times__insert,axiom,
    ! [A: b,A4: set_b,B: option7520157102916957007_c_nat,B5: set_op6398572775360143365_c_nat] :
      ( ( produc1892124643397453190_c_nat @ ( insert_b @ A @ A4 )
        @ ^ [Uu: b] : ( insert1278659629283105151_c_nat @ B @ B5 ) )
      = ( insert8034245793825879341_c_nat @ ( produc5716802255202478839_c_nat @ A @ B )
        @ ( sup_su1518086237216066783_c_nat
          @ ( produc1892124643397453190_c_nat @ A4
            @ ^ [Uu: b] : ( insert1278659629283105151_c_nat @ B @ B5 ) )
          @ ( produc1892124643397453190_c_nat @ ( insert_b @ A @ A4 )
            @ ^ [Uu: b] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_465_insert__Times__insert,axiom,
    ! [A: c,A4: set_c,B: nat,B5: set_nat] :
      ( ( product_Sigma_c_nat @ ( insert_c @ A @ A4 )
        @ ^ [Uu: c] : ( insert_nat @ B @ B5 ) )
      = ( insert1302120044651762863_c_nat @ ( product_Pair_c_nat @ A @ B )
        @ ( sup_su2428977078917480673_c_nat
          @ ( product_Sigma_c_nat @ A4
            @ ^ [Uu: c] : ( insert_nat @ B @ B5 ) )
          @ ( product_Sigma_c_nat @ ( insert_c @ A @ A4 )
            @ ^ [Uu: c] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_466_range__fst,axiom,
    ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ top_to689330392550168593_c_nat )
    = top_top_set_b ) ).

% range_fst
thf(fact_467_range__fst,axiom,
    ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ top_to7569790511745352472od_b_c )
    = top_top_set_b ) ).

% range_fst
thf(fact_468_range__fst,axiom,
    ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ top_to7683274595066931299_c_nat )
    = top_top_set_b ) ).

% range_fst
thf(fact_469_fst__map__prod,axiom,
    ! [F: b > b,G: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,X4: produc6689720525226317995_c_nat] :
      ( ( produc6033490199168946105_c_nat @ ( produc1006641225684822322_c_nat @ F @ G @ X4 ) )
      = ( F @ ( produc6033490199168946105_c_nat @ X4 ) ) ) ).

% fst_map_prod
thf(fact_470_fst__map__prod,axiom,
    ! [F: b > b,G: c > produc4862256710654508797_c_nat,X4: product_prod_b_c] :
      ( ( produc6033490199168946105_c_nat @ ( produc2579347470806298311_c_nat @ F @ G @ X4 ) )
      = ( F @ ( product_fst_b_c @ X4 ) ) ) ).

% fst_map_prod
thf(fact_471_fst__map__prod,axiom,
    ! [F: b > b,G: option7520157102916957007_c_nat > produc4862256710654508797_c_nat,X4: produc4862256710654508797_c_nat] :
      ( ( produc6033490199168946105_c_nat @ ( produc4824211994881367428_c_nat @ F @ G @ X4 ) )
      = ( F @ ( produc8816143915943518219_c_nat @ X4 ) ) ) ).

% fst_map_prod
thf(fact_472_fst__map__prod,axiom,
    ! [F: b > b,G: produc4862256710654508797_c_nat > c,X4: produc6689720525226317995_c_nat] :
      ( ( product_fst_b_c @ ( produc2181303955692932747_nat_c @ F @ G @ X4 ) )
      = ( F @ ( produc6033490199168946105_c_nat @ X4 ) ) ) ).

% fst_map_prod
thf(fact_473_fst__map__prod,axiom,
    ! [F: b > b,G: c > c,X4: product_prod_b_c] :
      ( ( product_fst_b_c @ ( produc5527967938102404790_b_c_c @ F @ G @ X4 ) )
      = ( F @ ( product_fst_b_c @ X4 ) ) ) ).

% fst_map_prod
thf(fact_474_fst__map__prod,axiom,
    ! [F: b > b,G: option7520157102916957007_c_nat > c,X4: produc4862256710654508797_c_nat] :
      ( ( product_fst_b_c @ ( produc3202252859625268153_nat_c @ F @ G @ X4 ) )
      = ( F @ ( produc8816143915943518219_c_nat @ X4 ) ) ) ).

% fst_map_prod
thf(fact_475_fst__map__prod,axiom,
    ! [F: b > b,G: produc4862256710654508797_c_nat > option7520157102916957007_c_nat,X4: produc6689720525226317995_c_nat] :
      ( ( produc8816143915943518219_c_nat @ ( produc2309332835414039172_c_nat @ F @ G @ X4 ) )
      = ( F @ ( produc6033490199168946105_c_nat @ X4 ) ) ) ).

% fst_map_prod
thf(fact_476_fst__map__prod,axiom,
    ! [F: b > b,G: c > option7520157102916957007_c_nat,X4: product_prod_b_c] :
      ( ( produc8816143915943518219_c_nat @ ( produc2445590922777410841_c_nat @ F @ G @ X4 ) )
      = ( F @ ( product_fst_b_c @ X4 ) ) ) ).

% fst_map_prod
thf(fact_477_fst__map__prod,axiom,
    ! [F: b > b,G: option7520157102916957007_c_nat > option7520157102916957007_c_nat,X4: produc4862256710654508797_c_nat] :
      ( ( produc8816143915943518219_c_nat @ ( produc3763450287754571990_c_nat @ F @ G @ X4 ) )
      = ( F @ ( produc8816143915943518219_c_nat @ X4 ) ) ) ).

% fst_map_prod
thf(fact_478_asym__onI,axiom,
    ! [A4: set_b,R2: set_Product_prod_b_b] :
      ( ! [X: b,Y: b] :
          ( ( member_b @ X @ A4 )
         => ( ( member_b @ Y @ A4 )
           => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ R2 )
             => ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y @ X ) @ R2 ) ) ) )
     => ( asym_on_b @ A4 @ R2 ) ) ).

% asym_onI
thf(fact_479_prod_Ocollapse,axiom,
    ! [Prod: product_prod_c_nat] :
      ( ( product_Pair_c_nat @ ( product_fst_c_nat @ Prod ) @ ( product_snd_c_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_480_prod_Ocollapse,axiom,
    ! [Prod: produc6689720525226317995_c_nat] :
      ( ( produc8868828497325608613_c_nat @ ( produc6033490199168946105_c_nat @ Prod ) @ ( produc6319817161620345591_c_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_481_prod_Ocollapse,axiom,
    ! [Prod: product_prod_b_c] :
      ( ( product_Pair_b_c @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_482_prod_Ocollapse,axiom,
    ! [Prod: produc4862256710654508797_c_nat] :
      ( ( produc5716802255202478839_c_nat @ ( produc8816143915943518219_c_nat @ Prod ) @ ( produc2435110264752509257_c_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_483_snd__swap,axiom,
    ! [X4: produc6689720525226317995_c_nat] :
      ( ( produc296059414264809275_nat_b @ ( produc1795597653127305477_c_nat @ X4 ) )
      = ( produc6033490199168946105_c_nat @ X4 ) ) ).

% snd_swap
thf(fact_484_snd__swap,axiom,
    ! [X4: product_prod_b_c] :
      ( ( product_snd_c_b @ ( product_swap_b_c @ X4 ) )
      = ( product_fst_b_c @ X4 ) ) ).

% snd_swap
thf(fact_485_snd__swap,axiom,
    ! [X4: produc4862256710654508797_c_nat] :
      ( ( produc6564488572305012073_nat_b @ ( produc8310569085468513623_c_nat @ X4 ) )
      = ( produc8816143915943518219_c_nat @ X4 ) ) ).

% snd_swap
thf(fact_486_fst__swap,axiom,
    ! [X4: produc3143399696777836727_nat_b] :
      ( ( produc6033490199168946105_c_nat @ ( produc4995211942626544969_nat_b @ X4 ) )
      = ( produc296059414264809275_nat_b @ X4 ) ) ).

% fst_swap
thf(fact_487_fst__swap,axiom,
    ! [X4: product_prod_c_b] :
      ( ( product_fst_b_c @ ( product_swap_c_b @ X4 ) )
      = ( product_snd_c_b @ X4 ) ) ).

% fst_swap
thf(fact_488_fst__swap,axiom,
    ! [X4: produc7545861912598698981_nat_b] :
      ( ( produc8816143915943518219_c_nat @ ( produc3216575356166240631_nat_b @ X4 ) )
      = ( produc6564488572305012073_nat_b @ X4 ) ) ).

% fst_swap
thf(fact_489_map__fst__map__ran,axiom,
    ! [F: b > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Al: list_P7417839048565863355_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( map_ra2537877954580484484_c_nat @ F @ Al ) )
      = ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) ) ).

% map_fst_map_ran
thf(fact_490_map__fst__map__ran,axiom,
    ! [F: b > c > produc4862256710654508797_c_nat,Al: list_P903359562653991662od_b_c] :
      ( ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( map_ra6584430756116811765_c_nat @ F @ Al ) )
      = ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) ) ).

% map_fst_map_ran
thf(fact_491_map__fst__map__ran,axiom,
    ! [F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,Al: list_P8155366108056616461_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( map_ra5752783577693071318_c_nat @ F @ Al ) )
      = ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) ) ).

% map_fst_map_ran
thf(fact_492_map__fst__map__ran,axiom,
    ! [F: b > produc4862256710654508797_c_nat > c,Al: list_P7417839048565863355_c_nat] :
      ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( map_ra6186387241003446201_nat_c @ F @ Al ) )
      = ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) ) ).

% map_fst_map_ran
thf(fact_493_map__fst__map__ran,axiom,
    ! [F: b > c > c,Al: list_P903359562653991662od_b_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( map_ran_b_c_c @ F @ Al ) )
      = ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) ) ).

% map_fst_map_ran
thf(fact_494_map__fst__map__ran,axiom,
    ! [F: b > option7520157102916957007_c_nat > c,Al: list_P8155366108056616461_c_nat] :
      ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( map_ra3617418671075910375_nat_c @ F @ Al ) )
      = ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) ) ).

% map_fst_map_ran
thf(fact_495_map__fst__map__ran,axiom,
    ! [F: b > produc4862256710654508797_c_nat > option7520157102916957007_c_nat,Al: list_P7417839048565863355_c_nat] :
      ( ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( map_ra3237904418225743062_c_nat @ F @ Al ) )
      = ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) ) ).

% map_fst_map_ran
thf(fact_496_map__fst__map__ran,axiom,
    ! [F: b > c > option7520157102916957007_c_nat,Al: list_P903359562653991662od_b_c] :
      ( ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( map_ra2860756734228053063_c_nat @ F @ Al ) )
      = ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) ) ).

% map_fst_map_ran
thf(fact_497_map__fst__map__ran,axiom,
    ! [F: b > option7520157102916957007_c_nat > option7520157102916957007_c_nat,Al: list_P8155366108056616461_c_nat] :
      ( ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( map_ra1379142644618337576_c_nat @ F @ Al ) )
      = ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) ) ).

% map_fst_map_ran
thf(fact_498_fst__image__times,axiom,
    ! [B5: set_Pr8806432033423503795_c_nat,A4: set_b] :
      ( ( ( B5 = bot_bo7474904860610844231_c_nat )
       => ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat
            @ ( produc237220987571061300_c_nat @ A4
              @ ^ [Uu: b] : B5 ) )
          = bot_bot_set_b ) )
      & ( ( B5 != bot_bo7474904860610844231_c_nat )
       => ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat
            @ ( produc237220987571061300_c_nat @ A4
              @ ^ [Uu: b] : B5 ) )
          = A4 ) ) ) ).

% fst_image_times
thf(fact_499_fst__image__times,axiom,
    ! [B5: set_c,A4: set_b] :
      ( ( ( B5 = bot_bot_set_c )
       => ( ( image_7762865866877615260_b_c_b @ product_fst_b_c
            @ ( product_Sigma_b_c @ A4
              @ ^ [Uu: b] : B5 ) )
          = bot_bot_set_b ) )
      & ( ( B5 != bot_bot_set_c )
       => ( ( image_7762865866877615260_b_c_b @ product_fst_b_c
            @ ( product_Sigma_b_c @ A4
              @ ^ [Uu: b] : B5 ) )
          = A4 ) ) ) ).

% fst_image_times
thf(fact_500_fst__image__times,axiom,
    ! [B5: set_op6398572775360143365_c_nat,A4: set_b] :
      ( ( ( B5 = bot_bo4013181551729047705_c_nat )
       => ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat
            @ ( produc1892124643397453190_c_nat @ A4
              @ ^ [Uu: b] : B5 ) )
          = bot_bot_set_b ) )
      & ( ( B5 != bot_bo4013181551729047705_c_nat )
       => ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat
            @ ( produc1892124643397453190_c_nat @ A4
              @ ^ [Uu: b] : B5 ) )
          = A4 ) ) ) ).

% fst_image_times
thf(fact_501_surjective__pairing,axiom,
    ! [T: product_prod_c_nat] :
      ( T
      = ( product_Pair_c_nat @ ( product_fst_c_nat @ T ) @ ( product_snd_c_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_502_surjective__pairing,axiom,
    ! [T: produc6689720525226317995_c_nat] :
      ( T
      = ( produc8868828497325608613_c_nat @ ( produc6033490199168946105_c_nat @ T ) @ ( produc6319817161620345591_c_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_503_surjective__pairing,axiom,
    ! [T: product_prod_b_c] :
      ( T
      = ( product_Pair_b_c @ ( product_fst_b_c @ T ) @ ( product_snd_b_c @ T ) ) ) ).

% surjective_pairing
thf(fact_504_surjective__pairing,axiom,
    ! [T: produc4862256710654508797_c_nat] :
      ( T
      = ( produc5716802255202478839_c_nat @ ( produc8816143915943518219_c_nat @ T ) @ ( produc2435110264752509257_c_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_505_conjI__realizer,axiom,
    ! [P2: c > $o,P: c,Q2: nat > $o,Q3: nat] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q3 )
       => ( ( P2 @ ( product_fst_c_nat @ ( product_Pair_c_nat @ P @ Q3 ) ) )
          & ( Q2 @ ( product_snd_c_nat @ ( product_Pair_c_nat @ P @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_506_conjI__realizer,axiom,
    ! [P2: b > $o,P: b,Q2: produc4862256710654508797_c_nat > $o,Q3: produc4862256710654508797_c_nat] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q3 )
       => ( ( P2 @ ( produc6033490199168946105_c_nat @ ( produc8868828497325608613_c_nat @ P @ Q3 ) ) )
          & ( Q2 @ ( produc6319817161620345591_c_nat @ ( produc8868828497325608613_c_nat @ P @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_507_conjI__realizer,axiom,
    ! [P2: b > $o,P: b,Q2: c > $o,Q3: c] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q3 )
       => ( ( P2 @ ( product_fst_b_c @ ( product_Pair_b_c @ P @ Q3 ) ) )
          & ( Q2 @ ( product_snd_b_c @ ( product_Pair_b_c @ P @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_508_conjI__realizer,axiom,
    ! [P2: b > $o,P: b,Q2: option7520157102916957007_c_nat > $o,Q3: option7520157102916957007_c_nat] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q3 )
       => ( ( P2 @ ( produc8816143915943518219_c_nat @ ( produc5716802255202478839_c_nat @ P @ Q3 ) ) )
          & ( Q2 @ ( produc2435110264752509257_c_nat @ ( produc5716802255202478839_c_nat @ P @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_509_exI__realizer,axiom,
    ! [P2: nat > c > $o,Y4: nat,X4: c] :
      ( ( P2 @ Y4 @ X4 )
     => ( P2 @ ( product_snd_c_nat @ ( product_Pair_c_nat @ X4 @ Y4 ) ) @ ( product_fst_c_nat @ ( product_Pair_c_nat @ X4 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_510_exI__realizer,axiom,
    ! [P2: produc4862256710654508797_c_nat > b > $o,Y4: produc4862256710654508797_c_nat,X4: b] :
      ( ( P2 @ Y4 @ X4 )
     => ( P2 @ ( produc6319817161620345591_c_nat @ ( produc8868828497325608613_c_nat @ X4 @ Y4 ) ) @ ( produc6033490199168946105_c_nat @ ( produc8868828497325608613_c_nat @ X4 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_511_exI__realizer,axiom,
    ! [P2: c > b > $o,Y4: c,X4: b] :
      ( ( P2 @ Y4 @ X4 )
     => ( P2 @ ( product_snd_b_c @ ( product_Pair_b_c @ X4 @ Y4 ) ) @ ( product_fst_b_c @ ( product_Pair_b_c @ X4 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_512_exI__realizer,axiom,
    ! [P2: option7520157102916957007_c_nat > b > $o,Y4: option7520157102916957007_c_nat,X4: b] :
      ( ( P2 @ Y4 @ X4 )
     => ( P2 @ ( produc2435110264752509257_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) ) @ ( produc8816143915943518219_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) ) ) ) ).

% exI_realizer
thf(fact_513_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_c_nat] :
      ( Prod
      = ( product_Pair_c_nat @ ( product_fst_c_nat @ Prod ) @ ( product_snd_c_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_514_prod_Oexhaust__sel,axiom,
    ! [Prod: produc6689720525226317995_c_nat] :
      ( Prod
      = ( produc8868828497325608613_c_nat @ ( produc6033490199168946105_c_nat @ Prod ) @ ( produc6319817161620345591_c_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_515_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_b_c] :
      ( Prod
      = ( product_Pair_b_c @ ( product_fst_b_c @ Prod ) @ ( product_snd_b_c @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_516_prod_Oexhaust__sel,axiom,
    ! [Prod: produc4862256710654508797_c_nat] :
      ( Prod
      = ( produc5716802255202478839_c_nat @ ( produc8816143915943518219_c_nat @ Prod ) @ ( produc2435110264752509257_c_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_517_prod_Oexpand,axiom,
    ! [Prod: produc6689720525226317995_c_nat,Prod2: produc6689720525226317995_c_nat] :
      ( ( ( ( produc6033490199168946105_c_nat @ Prod )
          = ( produc6033490199168946105_c_nat @ Prod2 ) )
        & ( ( produc6319817161620345591_c_nat @ Prod )
          = ( produc6319817161620345591_c_nat @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_518_prod_Oexpand,axiom,
    ! [Prod: product_prod_b_c,Prod2: product_prod_b_c] :
      ( ( ( ( product_fst_b_c @ Prod )
          = ( product_fst_b_c @ Prod2 ) )
        & ( ( product_snd_b_c @ Prod )
          = ( product_snd_b_c @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_519_prod_Oexpand,axiom,
    ! [Prod: produc4862256710654508797_c_nat,Prod2: produc4862256710654508797_c_nat] :
      ( ( ( ( produc8816143915943518219_c_nat @ Prod )
          = ( produc8816143915943518219_c_nat @ Prod2 ) )
        & ( ( produc2435110264752509257_c_nat @ Prod )
          = ( produc2435110264752509257_c_nat @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_520_prod__eqI,axiom,
    ! [P: produc6689720525226317995_c_nat,Q3: produc6689720525226317995_c_nat] :
      ( ( ( produc6033490199168946105_c_nat @ P )
        = ( produc6033490199168946105_c_nat @ Q3 ) )
     => ( ( ( produc6319817161620345591_c_nat @ P )
          = ( produc6319817161620345591_c_nat @ Q3 ) )
       => ( P = Q3 ) ) ) ).

% prod_eqI
thf(fact_521_prod__eqI,axiom,
    ! [P: product_prod_b_c,Q3: product_prod_b_c] :
      ( ( ( product_fst_b_c @ P )
        = ( product_fst_b_c @ Q3 ) )
     => ( ( ( product_snd_b_c @ P )
          = ( product_snd_b_c @ Q3 ) )
       => ( P = Q3 ) ) ) ).

% prod_eqI
thf(fact_522_prod__eqI,axiom,
    ! [P: produc4862256710654508797_c_nat,Q3: produc4862256710654508797_c_nat] :
      ( ( ( produc8816143915943518219_c_nat @ P )
        = ( produc8816143915943518219_c_nat @ Q3 ) )
     => ( ( ( produc2435110264752509257_c_nat @ P )
          = ( produc2435110264752509257_c_nat @ Q3 ) )
       => ( P = Q3 ) ) ) ).

% prod_eqI
thf(fact_523_exE__realizer_H,axiom,
    ! [P2: produc4862256710654508797_c_nat > b > $o,P: produc6689720525226317995_c_nat] :
      ( ( P2 @ ( produc6319817161620345591_c_nat @ P ) @ ( produc6033490199168946105_c_nat @ P ) )
     => ~ ! [X: b,Y: produc4862256710654508797_c_nat] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_524_exE__realizer_H,axiom,
    ! [P2: c > b > $o,P: product_prod_b_c] :
      ( ( P2 @ ( product_snd_b_c @ P ) @ ( product_fst_b_c @ P ) )
     => ~ ! [X: b,Y: c] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_525_exE__realizer_H,axiom,
    ! [P2: option7520157102916957007_c_nat > b > $o,P: produc4862256710654508797_c_nat] :
      ( ( P2 @ ( produc2435110264752509257_c_nat @ P ) @ ( produc8816143915943518219_c_nat @ P ) )
     => ~ ! [X: b,Y: option7520157102916957007_c_nat] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_526_prod__eq__iff,axiom,
    ( ( ^ [Y7: produc6689720525226317995_c_nat,Z2: produc6689720525226317995_c_nat] : ( Y7 = Z2 ) )
    = ( ^ [S4: produc6689720525226317995_c_nat,T2: produc6689720525226317995_c_nat] :
          ( ( ( produc6033490199168946105_c_nat @ S4 )
            = ( produc6033490199168946105_c_nat @ T2 ) )
          & ( ( produc6319817161620345591_c_nat @ S4 )
            = ( produc6319817161620345591_c_nat @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_527_prod__eq__iff,axiom,
    ( ( ^ [Y7: product_prod_b_c,Z2: product_prod_b_c] : ( Y7 = Z2 ) )
    = ( ^ [S4: product_prod_b_c,T2: product_prod_b_c] :
          ( ( ( product_fst_b_c @ S4 )
            = ( product_fst_b_c @ T2 ) )
          & ( ( product_snd_b_c @ S4 )
            = ( product_snd_b_c @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_528_prod__eq__iff,axiom,
    ( ( ^ [Y7: produc4862256710654508797_c_nat,Z2: produc4862256710654508797_c_nat] : ( Y7 = Z2 ) )
    = ( ^ [S4: produc4862256710654508797_c_nat,T2: produc4862256710654508797_c_nat] :
          ( ( ( produc8816143915943518219_c_nat @ S4 )
            = ( produc8816143915943518219_c_nat @ T2 ) )
          & ( ( produc2435110264752509257_c_nat @ S4 )
            = ( produc2435110264752509257_c_nat @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_529_Un__def,axiom,
    ( sup_sup_set_b
    = ( ^ [A6: set_b,B6: set_b] :
          ( collect_b
          @ ^ [X3: b] :
              ( ( member_b @ X3 @ A6 )
              | ( member_b @ X3 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_530_mem__Times__iff,axiom,
    ! [X4: product_prod_b_b,A4: set_b,B5: set_b] :
      ( ( member7862447936710763792od_b_b @ X4
        @ ( product_Sigma_b_b @ A4
          @ ^ [Uu: b] : B5 ) )
      = ( ( member_b @ ( product_fst_b_b @ X4 ) @ A4 )
        & ( member_b @ ( product_snd_b_b @ X4 ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_531_mem__Times__iff,axiom,
    ! [X4: produc6689720525226317995_c_nat,A4: set_b,B5: set_Pr8806432033423503795_c_nat] :
      ( ( member1522478204950355394_c_nat @ X4
        @ ( produc237220987571061300_c_nat @ A4
          @ ^ [Uu: b] : B5 ) )
      = ( ( member_b @ ( produc6033490199168946105_c_nat @ X4 ) @ A4 )
        & ( member7562873241046315796_c_nat @ ( produc6319817161620345591_c_nat @ X4 ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_532_mem__Times__iff,axiom,
    ! [X4: product_prod_b_c,A4: set_b,B5: set_c] :
      ( ( member7862447941013992593od_b_c @ X4
        @ ( product_Sigma_b_c @ A4
          @ ^ [Uu: b] : B5 ) )
      = ( ( member_b @ ( product_fst_b_c @ X4 ) @ A4 )
        & ( member_c @ ( product_snd_b_c @ X4 ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_533_mem__Times__iff,axiom,
    ! [X4: produc4862256710654508797_c_nat,A4: set_b,B5: set_op6398572775360143365_c_nat] :
      ( ( member7562873241046315796_c_nat @ X4
        @ ( produc1892124643397453190_c_nat @ A4
          @ ^ [Uu: b] : B5 ) )
      = ( ( member_b @ ( produc8816143915943518219_c_nat @ X4 ) @ A4 )
        & ( member3208607818295432806_c_nat @ ( produc2435110264752509257_c_nat @ X4 ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_534_case__prod__unfold,axiom,
    ( produc3722540595886809633_c_nat
    = ( ^ [C3: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,P5: produc4862256710654508797_c_nat] : ( C3 @ ( produc8816143915943518219_c_nat @ P5 ) @ ( produc2435110264752509257_c_nat @ P5 ) ) ) ) ).

% case_prod_unfold
thf(fact_535_case__prod__unfold,axiom,
    ( produc3293230613241633187_c_nat
    = ( ^ [C3: c > nat > produc4862256710654508797_c_nat,P5: product_prod_c_nat] : ( C3 @ ( product_fst_c_nat @ P5 ) @ ( product_snd_c_nat @ P5 ) ) ) ) ).

% case_prod_unfold
thf(fact_536_case__prod__beta_H,axiom,
    ( produc3722540595886809633_c_nat
    = ( ^ [F3: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,X3: produc4862256710654508797_c_nat] : ( F3 @ ( produc8816143915943518219_c_nat @ X3 ) @ ( produc2435110264752509257_c_nat @ X3 ) ) ) ) ).

% case_prod_beta'
thf(fact_537_case__prod__beta_H,axiom,
    ( produc3293230613241633187_c_nat
    = ( ^ [F3: c > nat > produc4862256710654508797_c_nat,X3: product_prod_c_nat] : ( F3 @ ( product_fst_c_nat @ X3 ) @ ( product_snd_c_nat @ X3 ) ) ) ) ).

% case_prod_beta'
thf(fact_538_exE__realizer,axiom,
    ! [P2: option7520157102916957007_c_nat > b > $o,P: produc4862256710654508797_c_nat,Q2: produc4862256710654508797_c_nat > $o,F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat] :
      ( ( P2 @ ( produc2435110264752509257_c_nat @ P ) @ ( produc8816143915943518219_c_nat @ P ) )
     => ( ! [X: b,Y: option7520157102916957007_c_nat] :
            ( ( P2 @ Y @ X )
           => ( Q2 @ ( F @ X @ Y ) ) )
       => ( Q2 @ ( produc3722540595886809633_c_nat @ F @ P ) ) ) ) ).

% exE_realizer
thf(fact_539_exE__realizer,axiom,
    ! [P2: nat > c > $o,P: product_prod_c_nat,Q2: produc4862256710654508797_c_nat > $o,F: c > nat > produc4862256710654508797_c_nat] :
      ( ( P2 @ ( product_snd_c_nat @ P ) @ ( product_fst_c_nat @ P ) )
     => ( ! [X: c,Y: nat] :
            ( ( P2 @ Y @ X )
           => ( Q2 @ ( F @ X @ Y ) ) )
       => ( Q2 @ ( produc3293230613241633187_c_nat @ F @ P ) ) ) ) ).

% exE_realizer
thf(fact_540_split__beta,axiom,
    ( produc3722540595886809633_c_nat
    = ( ^ [F3: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,Prod3: produc4862256710654508797_c_nat] : ( F3 @ ( produc8816143915943518219_c_nat @ Prod3 ) @ ( produc2435110264752509257_c_nat @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_541_split__beta,axiom,
    ( produc3293230613241633187_c_nat
    = ( ^ [F3: c > nat > produc4862256710654508797_c_nat,Prod3: product_prod_c_nat] : ( F3 @ ( product_fst_c_nat @ Prod3 ) @ ( product_snd_c_nat @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_542_case__prod__beta,axiom,
    ( produc3722540595886809633_c_nat
    = ( ^ [F3: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,P5: produc4862256710654508797_c_nat] : ( F3 @ ( produc8816143915943518219_c_nat @ P5 ) @ ( produc2435110264752509257_c_nat @ P5 ) ) ) ) ).

% case_prod_beta
thf(fact_543_case__prod__beta,axiom,
    ( produc3293230613241633187_c_nat
    = ( ^ [F3: c > nat > produc4862256710654508797_c_nat,P5: product_prod_c_nat] : ( F3 @ ( product_fst_c_nat @ P5 ) @ ( product_snd_c_nat @ P5 ) ) ) ) ).

% case_prod_beta
thf(fact_544_Product__Type_OCollect__case__prodD,axiom,
    ! [X4: produc6689720525226317995_c_nat,A4: b > produc4862256710654508797_c_nat > $o] :
      ( ( member1522478204950355394_c_nat @ X4 @ ( collec8595031719707015680_c_nat @ ( produc6484547981188837958_nat_o @ A4 ) ) )
     => ( A4 @ ( produc6033490199168946105_c_nat @ X4 ) @ ( produc6319817161620345591_c_nat @ X4 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_545_Product__Type_OCollect__case__prodD,axiom,
    ! [X4: product_prod_b_c,A4: b > c > $o] :
      ( ( member7862447941013992593od_b_c @ X4 @ ( collec548942224018234067od_b_c @ ( produc1873627588981547243_b_c_o @ A4 ) ) )
     => ( A4 @ ( product_fst_b_c @ X4 ) @ ( product_snd_b_c @ X4 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_546_Product__Type_OCollect__case__prodD,axiom,
    ! [X4: produc4862256710654508797_c_nat,A4: b > option7520157102916957007_c_nat > $o] :
      ( ( member7562873241046315796_c_nat @ X4 @ ( collec4170641529751027026_c_nat @ ( produc2544960137906461044_nat_o @ A4 ) ) )
     => ( A4 @ ( produc8816143915943518219_c_nat @ X4 ) @ ( produc2435110264752509257_c_nat @ X4 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_547_pair__list__eqI,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,Ys: list_P7417839048565863355_c_nat] :
      ( ( ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs )
        = ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Ys ) )
     => ( ( ( map_Pr4595408008933130094_c_nat @ produc6319817161620345591_c_nat @ Xs )
          = ( map_Pr4595408008933130094_c_nat @ produc6319817161620345591_c_nat @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_548_pair__list__eqI,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs )
        = ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Ys ) )
     => ( ( ( map_Pr8222292247188198876_b_c_c @ product_snd_b_c @ Xs )
          = ( map_Pr8222292247188198876_b_c_c @ product_snd_b_c @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_549_pair__list__eqI,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,Ys: list_P8155366108056616461_c_nat] :
      ( ( ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs )
        = ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Ys ) )
     => ( ( ( map_Pr8931211203828011794_c_nat @ produc2435110264752509257_c_nat @ Xs )
          = ( map_Pr8931211203828011794_c_nat @ produc2435110264752509257_c_nat @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_550_prod_Osplit__sel,axiom,
    ! [P2: produc4862256710654508797_c_nat > $o,F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,Prod: produc4862256710654508797_c_nat] :
      ( ( P2 @ ( produc3722540595886809633_c_nat @ F @ Prod ) )
      = ( ( Prod
          = ( produc5716802255202478839_c_nat @ ( produc8816143915943518219_c_nat @ Prod ) @ ( produc2435110264752509257_c_nat @ Prod ) ) )
       => ( P2 @ ( F @ ( produc8816143915943518219_c_nat @ Prod ) @ ( produc2435110264752509257_c_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_551_prod_Osplit__sel,axiom,
    ! [P2: produc4862256710654508797_c_nat > $o,F: c > nat > produc4862256710654508797_c_nat,Prod: product_prod_c_nat] :
      ( ( P2 @ ( produc3293230613241633187_c_nat @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_c_nat @ ( product_fst_c_nat @ Prod ) @ ( product_snd_c_nat @ Prod ) ) )
       => ( P2 @ ( F @ ( product_fst_c_nat @ Prod ) @ ( product_snd_c_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_552_prod_Osplit__sel__asm,axiom,
    ! [P2: produc4862256710654508797_c_nat > $o,F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,Prod: produc4862256710654508797_c_nat] :
      ( ( P2 @ ( produc3722540595886809633_c_nat @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc5716802255202478839_c_nat @ ( produc8816143915943518219_c_nat @ Prod ) @ ( produc2435110264752509257_c_nat @ Prod ) ) )
            & ~ ( P2 @ ( F @ ( produc8816143915943518219_c_nat @ Prod ) @ ( produc2435110264752509257_c_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_553_prod_Osplit__sel__asm,axiom,
    ! [P2: produc4862256710654508797_c_nat > $o,F: c > nat > produc4862256710654508797_c_nat,Prod: product_prod_c_nat] :
      ( ( P2 @ ( produc3293230613241633187_c_nat @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_c_nat @ ( product_fst_c_nat @ Prod ) @ ( product_snd_c_nat @ Prod ) ) )
            & ~ ( P2 @ ( F @ ( product_fst_c_nat @ Prod ) @ ( product_snd_c_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_554_zip__map__fst__snd,axiom,
    ! [Zs: list_P7417839048565863355_c_nat] :
      ( ( zip_b_1292320801234550661_c_nat @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Zs ) @ ( map_Pr4595408008933130094_c_nat @ produc6319817161620345591_c_nat @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_555_zip__map__fst__snd,axiom,
    ! [Zs: list_P903359562653991662od_b_c] :
      ( ( zip_b_c @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Zs ) @ ( map_Pr8222292247188198876_b_c_c @ product_snd_b_c @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_556_zip__map__fst__snd,axiom,
    ! [Zs: list_P8155366108056616461_c_nat] :
      ( ( zip_b_1039741285442616279_c_nat @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Zs ) @ ( map_Pr8931211203828011794_c_nat @ produc2435110264752509257_c_nat @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_557_prod_Oswap__def,axiom,
    ( produc3216575356166240631_nat_b
    = ( ^ [P5: produc7545861912598698981_nat_b] : ( produc5716802255202478839_c_nat @ ( produc6564488572305012073_nat_b @ P5 ) @ ( produc3722150186641245227_nat_b @ P5 ) ) ) ) ).

% prod.swap_def
thf(fact_558_prod_Oswap__def,axiom,
    ( product_swap_nat_c
    = ( ^ [P5: product_prod_nat_c] : ( product_Pair_c_nat @ ( product_snd_nat_c @ P5 ) @ ( product_fst_nat_c @ P5 ) ) ) ) ).

% prod.swap_def
thf(fact_559_prod_Oswap__def,axiom,
    ( produc1795597653127305477_c_nat
    = ( ^ [P5: produc6689720525226317995_c_nat] : ( produc2845070749970072297_nat_b @ ( produc6319817161620345591_c_nat @ P5 ) @ ( produc6033490199168946105_c_nat @ P5 ) ) ) ) ).

% prod.swap_def
thf(fact_560_prod_Oswap__def,axiom,
    ( product_swap_b_c
    = ( ^ [P5: product_prod_b_c] : ( product_Pair_c_b @ ( product_snd_b_c @ P5 ) @ ( product_fst_b_c @ P5 ) ) ) ) ).

% prod.swap_def
thf(fact_561_prod_Oswap__def,axiom,
    ( produc8310569085468513623_c_nat
    = ( ^ [P5: produc4862256710654508797_c_nat] : ( produc622808525900205847_nat_b @ ( produc2435110264752509257_c_nat @ P5 ) @ ( produc8816143915943518219_c_nat @ P5 ) ) ) ) ).

% prod.swap_def
thf(fact_562_snd__eqD,axiom,
    ! [X4: b,Y4: option7520157102916957007_c_nat,A: option7520157102916957007_c_nat] :
      ( ( ( produc2435110264752509257_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_563_snd__eqD,axiom,
    ! [X4: c,Y4: nat,A: nat] :
      ( ( ( product_snd_c_nat @ ( product_Pair_c_nat @ X4 @ Y4 ) )
        = A )
     => ( Y4 = A ) ) ).

% snd_eqD
thf(fact_564_snd__conv,axiom,
    ! [X1: b,X2: option7520157102916957007_c_nat] :
      ( ( produc2435110264752509257_c_nat @ ( produc5716802255202478839_c_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_565_snd__conv,axiom,
    ! [X1: c,X2: nat] :
      ( ( product_snd_c_nat @ ( product_Pair_c_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_566_fst__eqD,axiom,
    ! [X4: c,Y4: nat,A: c] :
      ( ( ( product_fst_c_nat @ ( product_Pair_c_nat @ X4 @ Y4 ) )
        = A )
     => ( X4 = A ) ) ).

% fst_eqD
thf(fact_567_fst__eqD,axiom,
    ! [X4: b,Y4: produc4862256710654508797_c_nat,A: b] :
      ( ( ( produc6033490199168946105_c_nat @ ( produc8868828497325608613_c_nat @ X4 @ Y4 ) )
        = A )
     => ( X4 = A ) ) ).

% fst_eqD
thf(fact_568_fst__eqD,axiom,
    ! [X4: b,Y4: c,A: b] :
      ( ( ( product_fst_b_c @ ( product_Pair_b_c @ X4 @ Y4 ) )
        = A )
     => ( X4 = A ) ) ).

% fst_eqD
thf(fact_569_fst__eqD,axiom,
    ! [X4: b,Y4: option7520157102916957007_c_nat,A: b] :
      ( ( ( produc8816143915943518219_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) )
        = A )
     => ( X4 = A ) ) ).

% fst_eqD
thf(fact_570_fst__conv,axiom,
    ! [X1: c,X2: nat] :
      ( ( product_fst_c_nat @ ( product_Pair_c_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_571_fst__conv,axiom,
    ! [X1: b,X2: produc4862256710654508797_c_nat] :
      ( ( produc6033490199168946105_c_nat @ ( produc8868828497325608613_c_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_572_fst__conv,axiom,
    ! [X1: b,X2: c] :
      ( ( product_fst_b_c @ ( product_Pair_b_c @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_573_fst__conv,axiom,
    ! [X1: b,X2: option7520157102916957007_c_nat] :
      ( ( produc8816143915943518219_c_nat @ ( produc5716802255202478839_c_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_574_fst__image__Sigma,axiom,
    ! [A4: set_b,B5: b > set_Pr8806432033423503795_c_nat] :
      ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( produc237220987571061300_c_nat @ A4 @ B5 ) )
      = ( collect_b
        @ ^ [X3: b] :
            ( ( member_b @ X3 @ A4 )
            & ( ( B5 @ X3 )
             != bot_bo7474904860610844231_c_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_575_fst__image__Sigma,axiom,
    ! [A4: set_b,B5: b > set_c] :
      ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( product_Sigma_b_c @ A4 @ B5 ) )
      = ( collect_b
        @ ^ [X3: b] :
            ( ( member_b @ X3 @ A4 )
            & ( ( B5 @ X3 )
             != bot_bot_set_c ) ) ) ) ).

% fst_image_Sigma
thf(fact_576_fst__image__Sigma,axiom,
    ! [A4: set_b,B5: b > set_op6398572775360143365_c_nat] :
      ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( produc1892124643397453190_c_nat @ A4 @ B5 ) )
      = ( collect_b
        @ ^ [X3: b] :
            ( ( member_b @ X3 @ A4 )
            & ( ( B5 @ X3 )
             != bot_bo4013181551729047705_c_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_577_The__case__prod,axiom,
    ! [P2: b > produc4862256710654508797_c_nat > $o] :
      ( ( the_Pr1484704762575096932_c_nat @ ( produc6484547981188837958_nat_o @ P2 ) )
      = ( the_Pr1484704762575096932_c_nat
        @ ^ [Xy: produc6689720525226317995_c_nat] : ( P2 @ ( produc6033490199168946105_c_nat @ Xy ) @ ( produc6319817161620345591_c_nat @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_578_The__case__prod,axiom,
    ! [P2: b > c > $o] :
      ( ( the_Product_prod_b_c @ ( produc1873627588981547243_b_c_o @ P2 ) )
      = ( the_Product_prod_b_c
        @ ^ [Xy: product_prod_b_c] : ( P2 @ ( product_fst_b_c @ Xy ) @ ( product_snd_b_c @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_579_The__case__prod,axiom,
    ! [P2: b > option7520157102916957007_c_nat > $o] :
      ( ( the_Pr4418202073612014518_c_nat @ ( produc2544960137906461044_nat_o @ P2 ) )
      = ( the_Pr4418202073612014518_c_nat
        @ ^ [Xy: produc4862256710654508797_c_nat] : ( P2 @ ( produc8816143915943518219_c_nat @ Xy ) @ ( produc2435110264752509257_c_nat @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_580_asym__onD,axiom,
    ! [A4: set_b,R2: set_Product_prod_b_b,X4: b,Y4: b] :
      ( ( asym_on_b @ A4 @ R2 )
     => ( ( member_b @ X4 @ A4 )
       => ( ( member_b @ Y4 @ A4 )
         => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
           => ~ ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y4 @ X4 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_581_fst__def,axiom,
    ( produc6033490199168946105_c_nat
    = ( produc1702926192408969901_nat_b
      @ ^ [X12: b,X22: produc4862256710654508797_c_nat] : X12 ) ) ).

% fst_def
thf(fact_582_fst__def,axiom,
    ( product_fst_b_c
    = ( produc8837194378803474130_b_c_b
      @ ^ [X12: b,X22: c] : X12 ) ) ).

% fst_def
thf(fact_583_fst__def,axiom,
    ( produc8816143915943518219_c_nat
    = ( produc9009988077629141723_nat_b
      @ ^ [X12: b,X22: option7520157102916957007_c_nat] : X12 ) ) ).

% fst_def
thf(fact_584_mmap__keys__update,axiom,
    ! [K2: b,V4: produc4862256710654508797_c_nat,Kvs: list_P7417839048565863355_c_nat] :
      ( ( mmap_k3080892797436903101_c_nat @ ( mmap_u2662500248198004552_c_nat @ K2 @ V4 @ Kvs ) )
      = ( sup_sup_set_b @ ( mmap_k3080892797436903101_c_nat @ Kvs ) @ ( insert_b @ K2 @ bot_bot_set_b ) ) ) ).

% mmap_keys_update
thf(fact_585_mmap__keys__update,axiom,
    ! [K2: b,V4: c,Kvs: list_P903359562653991662od_b_c] :
      ( ( mmap_keys_b_c @ ( mmap_update_b_c @ K2 @ V4 @ Kvs ) )
      = ( sup_sup_set_b @ ( mmap_keys_b_c @ Kvs ) @ ( insert_b @ K2 @ bot_bot_set_b ) ) ) ).

% mmap_keys_update
thf(fact_586_these__empty__eq,axiom,
    ! [B5: set_op6736743094888940163_c_nat] :
      ( ( ( these_5226245170637726720_c_nat @ B5 )
        = bot_bo7474904860610844231_c_nat )
      = ( ( B5 = bot_bo6496785114943191319_c_nat )
        | ( B5
          = ( insert6554936906190869117_c_nat @ none_P8487441334512977628_c_nat @ bot_bo6496785114943191319_c_nat ) ) ) ) ).

% these_empty_eq
thf(fact_587_these__not__empty__eq,axiom,
    ! [B5: set_op6736743094888940163_c_nat] :
      ( ( ( these_5226245170637726720_c_nat @ B5 )
       != bot_bo7474904860610844231_c_nat )
      = ( ( B5 != bot_bo6496785114943191319_c_nat )
        & ( B5
         != ( insert6554936906190869117_c_nat @ none_P8487441334512977628_c_nat @ bot_bo6496785114943191319_c_nat ) ) ) ) ).

% these_not_empty_eq
thf(fact_588_mmap__map__fst,axiom,
    ! [F: b > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,M: list_P7417839048565863355_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( mmap_m881505119971688169_c_nat @ F @ M ) )
      = ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ M ) ) ).

% mmap_map_fst
thf(fact_589_mmap__map__fst,axiom,
    ! [F: b > c > produc4862256710654508797_c_nat,M: list_P903359562653991662od_b_c] :
      ( ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( mmap_m3921707120754670672_c_nat @ F @ M ) )
      = ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ M ) ) ).

% mmap_map_fst
thf(fact_590_mmap__map__fst,axiom,
    ! [F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,M: list_P8155366108056616461_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( mmap_m6323497282726167099_c_nat @ F @ M ) )
      = ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ M ) ) ).

% mmap_map_fst
thf(fact_591_mmap__map__fst,axiom,
    ! [F: b > produc4862256710654508797_c_nat > c,M: list_P7417839048565863355_c_nat] :
      ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( mmap_m3523663605641305108_nat_c @ F @ M ) )
      = ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ M ) ) ).

% mmap_map_fst
thf(fact_592_mmap__map__fst,axiom,
    ! [F: b > c > c,M: list_P903359562653991662od_b_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( mmap_map_b_c_c @ F @ M ) )
      = ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ M ) ) ).

% mmap_map_fst
thf(fact_593_mmap__map__fst,axiom,
    ! [F: b > option7520157102916957007_c_nat > c,M: list_P8155366108056616461_c_nat] :
      ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( mmap_m309430304289022530_nat_c @ F @ M ) )
      = ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ M ) ) ).

% mmap_map_fst
thf(fact_594_mmap__map__fst,axiom,
    ! [F: b > produc4862256710654508797_c_nat > option7520157102916957007_c_nat,M: list_P7417839048565863355_c_nat] :
      ( ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( mmap_m3808618123258838843_c_nat @ F @ M ) )
      = ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ M ) ) ).

% mmap_map_fst
thf(fact_595_mmap__map__fst,axiom,
    ! [F: b > c > option7520157102916957007_c_nat,M: list_P903359562653991662od_b_c] :
      ( ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( mmap_m8776140404295941026_c_nat @ F @ M ) )
      = ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ M ) ) ).

% mmap_map_fst
thf(fact_596_mmap__map__fst,axiom,
    ! [F: b > option7520157102916957007_c_nat > option7520157102916957007_c_nat,M: list_P8155366108056616461_c_nat] :
      ( ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( mmap_m4119996586642187405_c_nat @ F @ M ) )
      = ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ M ) ) ).

% mmap_map_fst
thf(fact_597_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: c > nat > $o,X4: c,Y4: nat,A: product_prod_c_nat] :
      ( ( P2 @ X4 @ Y4 )
     => ( ( A
          = ( product_Pair_c_nat @ X4 @ Y4 ) )
       => ( P2 @ ( product_fst_c_nat @ A ) @ ( product_snd_c_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_598_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: b > produc4862256710654508797_c_nat > $o,X4: b,Y4: produc4862256710654508797_c_nat,A: produc6689720525226317995_c_nat] :
      ( ( P2 @ X4 @ Y4 )
     => ( ( A
          = ( produc8868828497325608613_c_nat @ X4 @ Y4 ) )
       => ( P2 @ ( produc6033490199168946105_c_nat @ A ) @ ( produc6319817161620345591_c_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_599_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: b > c > $o,X4: b,Y4: c,A: product_prod_b_c] :
      ( ( P2 @ X4 @ Y4 )
     => ( ( A
          = ( product_Pair_b_c @ X4 @ Y4 ) )
       => ( P2 @ ( product_fst_b_c @ A ) @ ( product_snd_b_c @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_600_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: b > option7520157102916957007_c_nat > $o,X4: b,Y4: option7520157102916957007_c_nat,A: produc4862256710654508797_c_nat] :
      ( ( P2 @ X4 @ Y4 )
     => ( ( A
          = ( produc5716802255202478839_c_nat @ X4 @ Y4 ) )
       => ( P2 @ ( produc8816143915943518219_c_nat @ A ) @ ( produc2435110264752509257_c_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_601_mmap__combine__lookup,axiom,
    ! [R2: list_P7417839048565863355_c_nat,K2: b,Z: b,V4: produc4862256710654508797_c_nat,C: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ R2 ) )
     => ( ( ( K2 = Z )
         => ( ( mmap_l5026999719965937591_c_nat @ ( mmap_c5208716839347718056_c_nat @ K2 @ V4 @ C @ R2 ) @ Z )
            = ( case_o2363692632768546263_c_nat @ ( some_P2720002978653898840_c_nat @ V4 )
              @ ^ [V5: produc4862256710654508797_c_nat] : ( some_P2720002978653898840_c_nat @ ( C @ V5 @ V4 ) )
              @ ( mmap_l5026999719965937591_c_nat @ R2 @ K2 ) ) ) )
        & ( ( K2 != Z )
         => ( ( mmap_l5026999719965937591_c_nat @ ( mmap_c5208716839347718056_c_nat @ K2 @ V4 @ C @ R2 ) @ Z )
            = ( mmap_l5026999719965937591_c_nat @ R2 @ Z ) ) ) ) ) ).

% mmap_combine_lookup
thf(fact_602_mmap__combine__lookup,axiom,
    ! [R2: list_P903359562653991662od_b_c,K2: b,Z: b,V4: c,C: c > c > c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ R2 ) )
     => ( ( ( K2 = Z )
         => ( ( mmap_lookup_b_c @ ( mmap_combine_b_c @ K2 @ V4 @ C @ R2 ) @ Z )
            = ( case_o1877681393294459589on_c_c @ ( some_c @ V4 )
              @ ^ [V5: c] : ( some_c @ ( C @ V5 @ V4 ) )
              @ ( mmap_lookup_b_c @ R2 @ K2 ) ) ) )
        & ( ( K2 != Z )
         => ( ( mmap_lookup_b_c @ ( mmap_combine_b_c @ K2 @ V4 @ C @ R2 ) @ Z )
            = ( mmap_lookup_b_c @ R2 @ Z ) ) ) ) ) ).

% mmap_combine_lookup
thf(fact_603_mmap__combine__lookup,axiom,
    ! [R2: list_P8155366108056616461_c_nat,K2: b,Z: b,V4: option7520157102916957007_c_nat,C: option7520157102916957007_c_nat > option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ R2 ) )
     => ( ( ( K2 = Z )
         => ( ( mmap_l1578203946870656521_c_nat @ ( mmap_c6869753822046002938_c_nat @ K2 @ V4 @ C @ R2 ) @ Z )
            = ( case_o7010352838590118779_c_nat @ ( some_o7552056839342673834_c_nat @ V4 )
              @ ^ [V5: option7520157102916957007_c_nat] : ( some_o7552056839342673834_c_nat @ ( C @ V5 @ V4 ) )
              @ ( mmap_l1578203946870656521_c_nat @ R2 @ K2 ) ) ) )
        & ( ( K2 != Z )
         => ( ( mmap_l1578203946870656521_c_nat @ ( mmap_c6869753822046002938_c_nat @ K2 @ V4 @ C @ R2 ) @ Z )
            = ( mmap_l1578203946870656521_c_nat @ R2 @ Z ) ) ) ) ) ).

% mmap_combine_lookup
thf(fact_604_sndI,axiom,
    ! [X4: produc4862256710654508797_c_nat,Y4: b,Z: option7520157102916957007_c_nat] :
      ( ( X4
        = ( produc5716802255202478839_c_nat @ Y4 @ Z ) )
     => ( ( produc2435110264752509257_c_nat @ X4 )
        = Z ) ) ).

% sndI
thf(fact_605_sndI,axiom,
    ! [X4: product_prod_c_nat,Y4: c,Z: nat] :
      ( ( X4
        = ( product_Pair_c_nat @ Y4 @ Z ) )
     => ( ( product_snd_c_nat @ X4 )
        = Z ) ) ).

% sndI
thf(fact_606_eq__snd__iff,axiom,
    ! [B: option7520157102916957007_c_nat,P: produc4862256710654508797_c_nat] :
      ( ( B
        = ( produc2435110264752509257_c_nat @ P ) )
      = ( ? [A5: b] :
            ( P
            = ( produc5716802255202478839_c_nat @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_607_eq__snd__iff,axiom,
    ! [B: nat,P: product_prod_c_nat] :
      ( ( B
        = ( product_snd_c_nat @ P ) )
      = ( ? [A5: c] :
            ( P
            = ( product_Pair_c_nat @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_608_eq__fst__iff,axiom,
    ! [A: c,P: product_prod_c_nat] :
      ( ( A
        = ( product_fst_c_nat @ P ) )
      = ( ? [B4: nat] :
            ( P
            = ( product_Pair_c_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_609_eq__fst__iff,axiom,
    ! [A: b,P: produc6689720525226317995_c_nat] :
      ( ( A
        = ( produc6033490199168946105_c_nat @ P ) )
      = ( ? [B4: produc4862256710654508797_c_nat] :
            ( P
            = ( produc8868828497325608613_c_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_610_eq__fst__iff,axiom,
    ! [A: b,P: product_prod_b_c] :
      ( ( A
        = ( product_fst_b_c @ P ) )
      = ( ? [B4: c] :
            ( P
            = ( product_Pair_b_c @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_611_eq__fst__iff,axiom,
    ! [A: b,P: produc4862256710654508797_c_nat] :
      ( ( A
        = ( produc8816143915943518219_c_nat @ P ) )
      = ( ? [B4: option7520157102916957007_c_nat] :
            ( P
            = ( produc5716802255202478839_c_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_612_distinct__before_I1_J,axiom,
    distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ s3 ) ).

% distinct_before(1)
thf(fact_613_sup__set__def,axiom,
    ( sup_sup_set_b
    = ( ^ [A6: set_b,B6: set_b] :
          ( collect_b
          @ ( sup_sup_b_o
            @ ^ [X3: b] : ( member_b @ X3 @ A6 )
            @ ^ [X3: b] : ( member_b @ X3 @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_614_sup__Un__eq,axiom,
    ! [R: set_b,S: set_b] :
      ( ( sup_sup_b_o
        @ ^ [X3: b] : ( member_b @ X3 @ R )
        @ ^ [X3: b] : ( member_b @ X3 @ S ) )
      = ( ^ [X3: b] : ( member_b @ X3 @ ( sup_sup_set_b @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_615_sup__Un__eq2,axiom,
    ! [R: set_Pr8806432033423503795_c_nat,S: set_Pr8806432033423503795_c_nat] :
      ( ( sup_su2667635397647816210_nat_o
        @ ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X3 @ Y3 ) @ R )
        @ ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X3 @ Y3 ) @ S ) )
      = ( ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X3 @ Y3 ) @ ( sup_su1518086237216066783_c_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_616_sup__Un__eq2,axiom,
    ! [R: set_Pr6903500605879609269_c_nat,S: set_Pr6903500605879609269_c_nat] :
      ( ( sup_sup_c_nat_o
        @ ^ [X3: c,Y3: nat] : ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X3 @ Y3 ) @ R )
        @ ^ [X3: c,Y3: nat] : ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X3 @ Y3 ) @ S ) )
      = ( ^ [X3: c,Y3: nat] : ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X3 @ Y3 ) @ ( sup_su2428977078917480673_c_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_617_distinct__map__ran,axiom,
    ! [Al: list_P7417839048565863355_c_nat,F: b > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( map_ra2537877954580484484_c_nat @ F @ Al ) ) ) ) ).

% distinct_map_ran
thf(fact_618_distinct__map__ran,axiom,
    ! [Al: list_P7417839048565863355_c_nat,F: b > produc4862256710654508797_c_nat > c] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( map_ra6186387241003446201_nat_c @ F @ Al ) ) ) ) ).

% distinct_map_ran
thf(fact_619_distinct__map__ran,axiom,
    ! [Al: list_P7417839048565863355_c_nat,F: b > produc4862256710654508797_c_nat > option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) )
     => ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( map_ra3237904418225743062_c_nat @ F @ Al ) ) ) ) ).

% distinct_map_ran
thf(fact_620_distinct__map__ran,axiom,
    ! [Al: list_P903359562653991662od_b_c,F: b > c > produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( map_ra6584430756116811765_c_nat @ F @ Al ) ) ) ) ).

% distinct_map_ran
thf(fact_621_distinct__map__ran,axiom,
    ! [Al: list_P903359562653991662od_b_c,F: b > c > c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( map_ran_b_c_c @ F @ Al ) ) ) ) ).

% distinct_map_ran
thf(fact_622_distinct__map__ran,axiom,
    ! [Al: list_P903359562653991662od_b_c,F: b > c > option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) )
     => ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( map_ra2860756734228053063_c_nat @ F @ Al ) ) ) ) ).

% distinct_map_ran
thf(fact_623_distinct__map__ran,axiom,
    ! [Al: list_P8155366108056616461_c_nat,F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( map_ra5752783577693071318_c_nat @ F @ Al ) ) ) ) ).

% distinct_map_ran
thf(fact_624_distinct__map__ran,axiom,
    ! [Al: list_P8155366108056616461_c_nat,F: b > option7520157102916957007_c_nat > c] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( map_ra3617418671075910375_nat_c @ F @ Al ) ) ) ) ).

% distinct_map_ran
thf(fact_625_distinct__map__ran,axiom,
    ! [Al: list_P8155366108056616461_c_nat,F: b > option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) )
     => ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( map_ra1379142644618337576_c_nat @ F @ Al ) ) ) ) ).

% distinct_map_ran
thf(fact_626_mmap__update__distinct,axiom,
    ! [M: list_P7417839048565863355_c_nat,K2: b,V4: produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ M ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( mmap_u2662500248198004552_c_nat @ K2 @ V4 @ M ) ) ) ) ).

% mmap_update_distinct
thf(fact_627_mmap__update__distinct,axiom,
    ! [M: list_P903359562653991662od_b_c,K2: b,V4: c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ M ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( mmap_update_b_c @ K2 @ V4 @ M ) ) ) ) ).

% mmap_update_distinct
thf(fact_628_mmap__update__distinct,axiom,
    ! [M: list_P8155366108056616461_c_nat,K2: b,V4: option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ M ) )
     => ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( mmap_u5371215486469568154_c_nat @ K2 @ V4 @ M ) ) ) ) ).

% mmap_update_distinct
thf(fact_629_fstI,axiom,
    ! [X4: product_prod_c_nat,Y4: c,Z: nat] :
      ( ( X4
        = ( product_Pair_c_nat @ Y4 @ Z ) )
     => ( ( product_fst_c_nat @ X4 )
        = Y4 ) ) ).

% fstI
thf(fact_630_fstI,axiom,
    ! [X4: produc6689720525226317995_c_nat,Y4: b,Z: produc4862256710654508797_c_nat] :
      ( ( X4
        = ( produc8868828497325608613_c_nat @ Y4 @ Z ) )
     => ( ( produc6033490199168946105_c_nat @ X4 )
        = Y4 ) ) ).

% fstI
thf(fact_631_fstI,axiom,
    ! [X4: product_prod_b_c,Y4: b,Z: c] :
      ( ( X4
        = ( product_Pair_b_c @ Y4 @ Z ) )
     => ( ( product_fst_b_c @ X4 )
        = Y4 ) ) ).

% fstI
thf(fact_632_fstI,axiom,
    ! [X4: produc4862256710654508797_c_nat,Y4: b,Z: option7520157102916957007_c_nat] :
      ( ( X4
        = ( produc5716802255202478839_c_nat @ Y4 @ Z ) )
     => ( ( produc8816143915943518219_c_nat @ X4 )
        = Y4 ) ) ).

% fstI
thf(fact_633_valid__before_I6_J,axiom,
    distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ e2 ) ).

% valid_before(6)
thf(fact_634_fold__sup__distinct,axiom,
    ! [M: list_P903359562653991662od_b_c,F: b > b] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ M ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( fold_sup_b_c @ M @ F ) ) ) ) ).

% fold_sup_distinct
thf(fact_635_distinct__update__with__aux,axiom,
    ! [V4: produc4862256710654508797_c_nat,K2: b,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Ps: list_P7417839048565863355_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( update5508768958394535230_nat_b @ V4 @ K2 @ F @ Ps ) ) )
      = ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Ps ) ) ) ).

% distinct_update_with_aux
thf(fact_636_distinct__update__with__aux,axiom,
    ! [V4: c,K2: b,F: c > c,Ps: list_P903359562653991662od_b_c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( update_with_aux_c_b @ V4 @ K2 @ F @ Ps ) ) )
      = ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Ps ) ) ) ).

% distinct_update_with_aux
thf(fact_637_distinct__update__with__aux,axiom,
    ! [V4: option7520157102916957007_c_nat,K2: b,F: option7520157102916957007_c_nat > option7520157102916957007_c_nat,Ps: list_P8155366108056616461_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( update6007670785845294700_nat_b @ V4 @ K2 @ F @ Ps ) ) )
      = ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Ps ) ) ) ).

% distinct_update_with_aux
thf(fact_638_lookup__adv__d__Some,axiom,
    ! [Step: b > option7520157102916957007_c_nat > b,St: mappin7688773591249790157_nat_b,S2: list_P7417839048565863355_c_nat,I2: nat,Bs: option7520157102916957007_c_nat,X4: b,Z: b,S3: list_P7417839048565863355_c_nat,St2: mappin7688773591249790157_nat_b] :
      ( ! [Q4: b,Bs2: option7520157102916957007_c_nat] :
          ( case_option_o_b @ $true
          @ ( ^ [Y7: b,Z2: b] : ( Y7 = Z2 )
            @ ( Step @ Q4 @ Bs2 ) )
          @ ( lookup5460638230741284100_nat_b @ St @ ( produc5716802255202478839_c_nat @ Q4 @ Bs2 ) ) )
     => ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ S2 ) )
       => ( ( keys_i3309910835001723460_nat_c @ Step @ I2 @ Bs @ S2 )
         => ( ( member_b @ X4 @ ( mmap_k3080892797436903101_c_nat @ S2 ) )
           => ( ( ( Step @ X4 @ Bs )
                = Z )
             => ( ( ( adv_d_1868095517933120126_nat_c @ Step @ St @ I2 @ Bs @ S2 )
                  = ( produc8994345104281052025_nat_b @ S3 @ St2 ) )
               => ( ( mmap_l5026999719965937591_c_nat @ S3 @ Z )
                  = ( some_P2720002978653898840_c_nat @ ( drop_cur_b_c @ I2 @ ( the_Pr3263588667584626537_c_nat @ ( mmap_l5026999719965937591_c_nat @ S2 @ X4 ) ) ) ) ) ) ) ) ) ) ) ).

% lookup_adv_d_Some
thf(fact_639_mmap__combine__distinct__set,axiom,
    ! [R2: list_P7417839048565863355_c_nat,K2: b,V4: produc4862256710654508797_c_nat,C: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ R2 ) )
     => ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( mmap_c5208716839347718056_c_nat @ K2 @ V4 @ C @ R2 ) ) )
        & ( ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( mmap_c5208716839347718056_c_nat @ K2 @ V4 @ C @ R2 ) ) )
          = ( sup_sup_set_b @ ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ R2 ) ) @ ( insert_b @ K2 @ bot_bot_set_b ) ) ) ) ) ).

% mmap_combine_distinct_set
thf(fact_640_mmap__combine__distinct__set,axiom,
    ! [R2: list_P903359562653991662od_b_c,K2: b,V4: c,C: c > c > c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ R2 ) )
     => ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( mmap_combine_b_c @ K2 @ V4 @ C @ R2 ) ) )
        & ( ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( mmap_combine_b_c @ K2 @ V4 @ C @ R2 ) ) )
          = ( sup_sup_set_b @ ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ R2 ) ) @ ( insert_b @ K2 @ bot_bot_set_b ) ) ) ) ) ).

% mmap_combine_distinct_set
thf(fact_641_mmap__combine__distinct__set,axiom,
    ! [R2: list_P8155366108056616461_c_nat,K2: b,V4: option7520157102916957007_c_nat,C: option7520157102916957007_c_nat > option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ R2 ) )
     => ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( mmap_c6869753822046002938_c_nat @ K2 @ V4 @ C @ R2 ) ) )
        & ( ( set_b2 @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( mmap_c6869753822046002938_c_nat @ K2 @ V4 @ C @ R2 ) ) )
          = ( sup_sup_set_b @ ( set_b2 @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ R2 ) ) @ ( insert_b @ K2 @ bot_bot_set_b ) ) ) ) ) ).

% mmap_combine_distinct_set
thf(fact_642_e_H__def,axiom,
    ( e
    = ( mmap_update_b_c @ ( produc8816143915943518219_c_nat @ ( the_Pr3263588667584626537_c_nat @ ( mmap_l5026999719965937591_c_nat @ s3 @ init ) ) ) @ t @ e2 ) ) ).

% e'_def
thf(fact_643_map__eq__conv,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,G: produc6689720525226317995_c_nat > b] :
      ( ( ( map_Pr6200325787298368846_nat_b @ F @ Xs )
        = ( map_Pr6200325787298368846_nat_b @ G @ Xs ) )
      = ( ! [X3: produc6689720525226317995_c_nat] :
            ( ( member1522478204950355394_c_nat @ X3 @ ( set_Pr5807476769692288598_c_nat @ Xs ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) ) ) ) ).

% map_eq_conv
thf(fact_644_map__eq__conv,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,G: product_prod_b_c > b] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
        = ( map_Pr8222292247188198875_b_c_b @ G @ Xs ) )
      = ( ! [X3: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) ) ) ) ).

% map_eq_conv
thf(fact_645_list_Oset__map,axiom,
    ! [F: produc6689720525226317995_c_nat > b,V4: list_P7417839048565863355_c_nat] :
      ( ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ F @ V4 ) )
      = ( image_9118790927800078285_nat_b @ F @ ( set_Pr5807476769692288598_c_nat @ V4 ) ) ) ).

% list.set_map
thf(fact_646_list_Oset__map,axiom,
    ! [F: product_prod_b_c > b,V4: list_P903359562653991662od_b_c] :
      ( ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ F @ V4 ) )
      = ( image_7762865866877615260_b_c_b @ F @ ( set_Product_prod_b_c2 @ V4 ) ) ) ).

% list.set_map
thf(fact_647_set__zip__rightD,axiom,
    ! [X4: b,Y4: option7520157102916957007_c_nat,Xs: list_b,Ys: list_o9079336047566018143_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) @ ( set_Pr5735414202003984296_c_nat @ ( zip_b_1039741285442616279_c_nat @ Xs @ Ys ) ) )
     => ( member3208607818295432806_c_nat @ Y4 @ ( set_op620999267866781434_c_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_648_set__zip__rightD,axiom,
    ! [X4: c,Y4: nat,Xs: list_c,Ys: list_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X4 @ Y4 ) @ ( set_Pr3395872032415823530_c_nat @ ( zip_c_nat @ Xs @ Ys ) ) )
     => ( member_nat @ Y4 @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_649_set__zip__leftD,axiom,
    ! [X4: b,Y4: option7520157102916957007_c_nat,Xs: list_b,Ys: list_o9079336047566018143_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) @ ( set_Pr5735414202003984296_c_nat @ ( zip_b_1039741285442616279_c_nat @ Xs @ Ys ) ) )
     => ( member_b @ X4 @ ( set_b2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_650_set__zip__leftD,axiom,
    ! [X4: c,Y4: nat,Xs: list_c,Ys: list_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X4 @ Y4 ) @ ( set_Pr3395872032415823530_c_nat @ ( zip_c_nat @ Xs @ Ys ) ) )
     => ( member_c @ X4 @ ( set_c2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_651_in__set__zipE,axiom,
    ! [X4: b,Y4: b,Xs: list_b,Ys: list_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ ( set_Product_prod_b_b2 @ ( zip_b_b @ Xs @ Ys ) ) )
     => ~ ( ( member_b @ X4 @ ( set_b2 @ Xs ) )
         => ~ ( member_b @ Y4 @ ( set_b2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_652_in__set__zipE,axiom,
    ! [X4: b,Y4: option7520157102916957007_c_nat,Xs: list_b,Ys: list_o9079336047566018143_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) @ ( set_Pr5735414202003984296_c_nat @ ( zip_b_1039741285442616279_c_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_b @ X4 @ ( set_b2 @ Xs ) )
         => ~ ( member3208607818295432806_c_nat @ Y4 @ ( set_op620999267866781434_c_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_653_in__set__zipE,axiom,
    ! [X4: c,Y4: nat,Xs: list_c,Ys: list_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X4 @ Y4 ) @ ( set_Pr3395872032415823530_c_nat @ ( zip_c_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_c @ X4 @ ( set_c2 @ Xs ) )
         => ~ ( member_nat @ Y4 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_654_zip__same,axiom,
    ! [A: b,B: b,Xs: list_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( set_Product_prod_b_b2 @ ( zip_b_b @ Xs @ Xs ) ) )
      = ( ( member_b @ A @ ( set_b2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_655_ex__map__conv,axiom,
    ! [Ys: list_b,F: produc6689720525226317995_c_nat > b] :
      ( ( ? [Xs2: list_P7417839048565863355_c_nat] :
            ( Ys
            = ( map_Pr6200325787298368846_nat_b @ F @ Xs2 ) ) )
      = ( ! [X3: b] :
            ( ( member_b @ X3 @ ( set_b2 @ Ys ) )
           => ? [Y3: produc6689720525226317995_c_nat] :
                ( X3
                = ( F @ Y3 ) ) ) ) ) ).

% ex_map_conv
thf(fact_656_ex__map__conv,axiom,
    ! [Ys: list_b,F: product_prod_b_c > b] :
      ( ( ? [Xs2: list_P903359562653991662od_b_c] :
            ( Ys
            = ( map_Pr8222292247188198875_b_c_b @ F @ Xs2 ) ) )
      = ( ! [X3: b] :
            ( ( member_b @ X3 @ ( set_b2 @ Ys ) )
           => ? [Y3: product_prod_b_c] :
                ( X3
                = ( F @ Y3 ) ) ) ) ) ).

% ex_map_conv
thf(fact_657_map__cong,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,Ys: list_P7417839048565863355_c_nat,F: produc6689720525226317995_c_nat > b,G: produc6689720525226317995_c_nat > b] :
      ( ( Xs = Ys )
     => ( ! [X: produc6689720525226317995_c_nat] :
            ( ( member1522478204950355394_c_nat @ X @ ( set_Pr5807476769692288598_c_nat @ Ys ) )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( map_Pr6200325787298368846_nat_b @ F @ Xs )
          = ( map_Pr6200325787298368846_nat_b @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_658_map__cong,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c,F: product_prod_b_c > b,G: product_prod_b_c > b] :
      ( ( Xs = Ys )
     => ( ! [X: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Ys ) )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
          = ( map_Pr8222292247188198875_b_c_b @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_659_map__idI,axiom,
    ! [Xs: list_b,F: b > b] :
      ( ! [X: b] :
          ( ( member_b @ X @ ( set_b2 @ Xs ) )
         => ( ( F @ X )
            = X ) )
     => ( ( map_b_b @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_660_map__ext,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,F: produc6689720525226317995_c_nat > b,G: produc6689720525226317995_c_nat > b] :
      ( ! [X: produc6689720525226317995_c_nat] :
          ( ( member1522478204950355394_c_nat @ X @ ( set_Pr5807476769692288598_c_nat @ Xs ) )
         => ( ( F @ X )
            = ( G @ X ) ) )
     => ( ( map_Pr6200325787298368846_nat_b @ F @ Xs )
        = ( map_Pr6200325787298368846_nat_b @ G @ Xs ) ) ) ).

% map_ext
thf(fact_661_map__ext,axiom,
    ! [Xs: list_P903359562653991662od_b_c,F: product_prod_b_c > b,G: product_prod_b_c > b] :
      ( ! [X: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ X @ ( set_Product_prod_b_c2 @ Xs ) )
         => ( ( F @ X )
            = ( G @ X ) ) )
     => ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
        = ( map_Pr8222292247188198875_b_c_b @ G @ Xs ) ) ) ).

% map_ext
thf(fact_662_list_Omap__ident__strong,axiom,
    ! [T: list_b,F: b > b] :
      ( ! [Z3: b] :
          ( ( member_b @ Z3 @ ( set_b2 @ T ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_b_b @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_663_list_Oinj__map__strong,axiom,
    ! [X4: list_P7417839048565863355_c_nat,Xa: list_P7417839048565863355_c_nat,F: produc6689720525226317995_c_nat > b,Fa: produc6689720525226317995_c_nat > b] :
      ( ! [Z3: produc6689720525226317995_c_nat,Za: produc6689720525226317995_c_nat] :
          ( ( member1522478204950355394_c_nat @ Z3 @ ( set_Pr5807476769692288598_c_nat @ X4 ) )
         => ( ( member1522478204950355394_c_nat @ Za @ ( set_Pr5807476769692288598_c_nat @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_Pr6200325787298368846_nat_b @ F @ X4 )
          = ( map_Pr6200325787298368846_nat_b @ Fa @ Xa ) )
       => ( X4 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_664_list_Oinj__map__strong,axiom,
    ! [X4: list_P903359562653991662od_b_c,Xa: list_P903359562653991662od_b_c,F: product_prod_b_c > b,Fa: product_prod_b_c > b] :
      ( ! [Z3: product_prod_b_c,Za: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ Z3 @ ( set_Product_prod_b_c2 @ X4 ) )
         => ( ( member7862447941013992593od_b_c @ Za @ ( set_Product_prod_b_c2 @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_Pr8222292247188198875_b_c_b @ F @ X4 )
          = ( map_Pr8222292247188198875_b_c_b @ Fa @ Xa ) )
       => ( X4 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_665_list_Omap__cong0,axiom,
    ! [X4: list_P7417839048565863355_c_nat,F: produc6689720525226317995_c_nat > b,G: produc6689720525226317995_c_nat > b] :
      ( ! [Z3: produc6689720525226317995_c_nat] :
          ( ( member1522478204950355394_c_nat @ Z3 @ ( set_Pr5807476769692288598_c_nat @ X4 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_Pr6200325787298368846_nat_b @ F @ X4 )
        = ( map_Pr6200325787298368846_nat_b @ G @ X4 ) ) ) ).

% list.map_cong0
thf(fact_666_list_Omap__cong0,axiom,
    ! [X4: list_P903359562653991662od_b_c,F: product_prod_b_c > b,G: product_prod_b_c > b] :
      ( ! [Z3: product_prod_b_c] :
          ( ( member7862447941013992593od_b_c @ Z3 @ ( set_Product_prod_b_c2 @ X4 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_Pr8222292247188198875_b_c_b @ F @ X4 )
        = ( map_Pr8222292247188198875_b_c_b @ G @ X4 ) ) ) ).

% list.map_cong0
thf(fact_667_list_Omap__cong,axiom,
    ! [X4: list_P7417839048565863355_c_nat,Ya: list_P7417839048565863355_c_nat,F: produc6689720525226317995_c_nat > b,G: produc6689720525226317995_c_nat > b] :
      ( ( X4 = Ya )
     => ( ! [Z3: produc6689720525226317995_c_nat] :
            ( ( member1522478204950355394_c_nat @ Z3 @ ( set_Pr5807476769692288598_c_nat @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_Pr6200325787298368846_nat_b @ F @ X4 )
          = ( map_Pr6200325787298368846_nat_b @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_668_list_Omap__cong,axiom,
    ! [X4: list_P903359562653991662od_b_c,Ya: list_P903359562653991662od_b_c,F: product_prod_b_c > b,G: product_prod_b_c > b] :
      ( ( X4 = Ya )
     => ( ! [Z3: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ Z3 @ ( set_Product_prod_b_c2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_Pr8222292247188198875_b_c_b @ F @ X4 )
          = ( map_Pr8222292247188198875_b_c_b @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_669_image__set,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat] :
      ( ( image_9118790927800078285_nat_b @ F @ ( set_Pr5807476769692288598_c_nat @ Xs ) )
      = ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) ) ) ).

% image_set
thf(fact_670_image__set,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( image_7762865866877615260_b_c_b @ F @ ( set_Product_prod_b_c2 @ Xs ) )
      = ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) ) ) ).

% image_set
thf(fact_671_dom__update__with__aux,axiom,
    ! [V4: produc4862256710654508797_c_nat,K2: b,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Ps: list_P7417839048565863355_c_nat] :
      ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ ( update5508768958394535230_nat_b @ V4 @ K2 @ F @ Ps ) ) )
      = ( sup_sup_set_b @ ( insert_b @ K2 @ bot_bot_set_b ) @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Ps ) ) ) ) ).

% dom_update_with_aux
thf(fact_672_dom__update__with__aux,axiom,
    ! [V4: c,K2: b,F: c > c,Ps: list_P903359562653991662od_b_c] :
      ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( update_with_aux_c_b @ V4 @ K2 @ F @ Ps ) ) )
      = ( sup_sup_set_b @ ( insert_b @ K2 @ bot_bot_set_b ) @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Ps ) ) ) ) ).

% dom_update_with_aux
thf(fact_673_dom__update__with__aux,axiom,
    ! [V4: option7520157102916957007_c_nat,K2: b,F: option7520157102916957007_c_nat > option7520157102916957007_c_nat,Ps: list_P8155366108056616461_c_nat] :
      ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ ( update6007670785845294700_nat_b @ V4 @ K2 @ F @ Ps ) ) )
      = ( sup_sup_set_b @ ( insert_b @ K2 @ bot_bot_set_b ) @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Ps ) ) ) ) ).

% dom_update_with_aux
thf(fact_674_mmap__keys__def,axiom,
    ( mmap_k3080892797436903101_c_nat
    = ( ^ [Kvs2: list_P7417839048565863355_c_nat] : ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Kvs2 ) ) ) ) ).

% mmap_keys_def
thf(fact_675_mmap__keys__def,axiom,
    ( mmap_keys_b_c
    = ( ^ [Kvs2: list_P903359562653991662od_b_c] : ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Kvs2 ) ) ) ) ).

% mmap_keys_def
thf(fact_676_mmap__keys__def,axiom,
    ( mmap_k6910799909894206223_c_nat
    = ( ^ [Kvs2: list_P8155366108056616461_c_nat] : ( set_b2 @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Kvs2 ) ) ) ) ).

% mmap_keys_def
thf(fact_677_dom__map__ran,axiom,
    ! [F: b > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Al: list_P7417839048565863355_c_nat] :
      ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ ( map_ra2537877954580484484_c_nat @ F @ Al ) ) )
      = ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Al ) ) ) ).

% dom_map_ran
thf(fact_678_dom__map__ran,axiom,
    ! [F: b > c > produc4862256710654508797_c_nat,Al: list_P903359562653991662od_b_c] :
      ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ ( map_ra6584430756116811765_c_nat @ F @ Al ) ) )
      = ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Al ) ) ) ).

% dom_map_ran
thf(fact_679_dom__map__ran,axiom,
    ! [F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,Al: list_P8155366108056616461_c_nat] :
      ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ ( map_ra5752783577693071318_c_nat @ F @ Al ) ) )
      = ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Al ) ) ) ).

% dom_map_ran
thf(fact_680_dom__map__ran,axiom,
    ! [F: b > produc4862256710654508797_c_nat > c,Al: list_P7417839048565863355_c_nat] :
      ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( map_ra6186387241003446201_nat_c @ F @ Al ) ) )
      = ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Al ) ) ) ).

% dom_map_ran
thf(fact_681_dom__map__ran,axiom,
    ! [F: b > c > c,Al: list_P903359562653991662od_b_c] :
      ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( map_ran_b_c_c @ F @ Al ) ) )
      = ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Al ) ) ) ).

% dom_map_ran
thf(fact_682_dom__map__ran,axiom,
    ! [F: b > option7520157102916957007_c_nat > c,Al: list_P8155366108056616461_c_nat] :
      ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( map_ra3617418671075910375_nat_c @ F @ Al ) ) )
      = ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Al ) ) ) ).

% dom_map_ran
thf(fact_683_dom__map__ran,axiom,
    ! [F: b > produc4862256710654508797_c_nat > option7520157102916957007_c_nat,Al: list_P7417839048565863355_c_nat] :
      ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ ( map_ra3237904418225743062_c_nat @ F @ Al ) ) )
      = ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Al ) ) ) ).

% dom_map_ran
thf(fact_684_dom__map__ran,axiom,
    ! [F: b > c > option7520157102916957007_c_nat,Al: list_P903359562653991662od_b_c] :
      ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ ( map_ra2860756734228053063_c_nat @ F @ Al ) ) )
      = ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Al ) ) ) ).

% dom_map_ran
thf(fact_685_dom__map__ran,axiom,
    ! [F: b > option7520157102916957007_c_nat > option7520157102916957007_c_nat,Al: list_P8155366108056616461_c_nat] :
      ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ ( map_ra1379142644618337576_c_nat @ F @ Al ) ) )
      = ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Al ) ) ) ).

% dom_map_ran
thf(fact_686_adv__d__keys,axiom,
    ! [Step: b > option7520157102916957007_c_nat > b,St: mappin7688773591249790157_nat_b,S2: list_P7417839048565863355_c_nat,I2: nat,Bs: option7520157102916957007_c_nat,S3: list_P7417839048565863355_c_nat,St2: mappin7688773591249790157_nat_b] :
      ( ! [Q4: b,Bs2: option7520157102916957007_c_nat] :
          ( case_option_o_b @ $true
          @ ( ^ [Y7: b,Z2: b] : ( Y7 = Z2 )
            @ ( Step @ Q4 @ Bs2 ) )
          @ ( lookup5460638230741284100_nat_b @ St @ ( produc5716802255202478839_c_nat @ Q4 @ Bs2 ) ) )
     => ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ S2 ) )
       => ( ( ( adv_d_1868095517933120126_nat_c @ Step @ St @ I2 @ Bs @ S2 )
            = ( produc8994345104281052025_nat_b @ S3 @ St2 ) )
         => ( ( mmap_k3080892797436903101_c_nat @ S3 )
            = ( image_b_b
              @ ^ [Q5: b] : ( Step @ Q5 @ Bs )
              @ ( mmap_k3080892797436903101_c_nat @ S2 ) ) ) ) ) ) ).

% adv_d_keys
thf(fact_687_lookup__adv__d__None,axiom,
    ! [Step: b > option7520157102916957007_c_nat > b,St: mappin7688773591249790157_nat_b,S2: list_P7417839048565863355_c_nat,I2: nat,Bs: option7520157102916957007_c_nat,S3: list_P7417839048565863355_c_nat,St2: mappin7688773591249790157_nat_b,Z: b] :
      ( ! [Q4: b,Bs2: option7520157102916957007_c_nat] :
          ( case_option_o_b @ $true
          @ ( ^ [Y7: b,Z2: b] : ( Y7 = Z2 )
            @ ( Step @ Q4 @ Bs2 ) )
          @ ( lookup5460638230741284100_nat_b @ St @ ( produc5716802255202478839_c_nat @ Q4 @ Bs2 ) ) )
     => ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ S2 ) )
       => ( ( ( adv_d_1868095517933120126_nat_c @ Step @ St @ I2 @ Bs @ S2 )
            = ( produc8994345104281052025_nat_b @ S3 @ St2 ) )
         => ( ( ( collect_b
                @ ^ [X3: b] :
                    ( ( member_b @ X3 @ ( mmap_k3080892797436903101_c_nat @ S2 ) )
                    & ( ( Step @ X3 @ Bs )
                      = Z ) ) )
              = bot_bot_set_b )
           => ( ( mmap_l5026999719965937591_c_nat @ S3 @ Z )
              = none_P8487441334512977628_c_nat ) ) ) ) ) ).

% lookup_adv_d_None
thf(fact_688_mmap__lookup__distinct,axiom,
    ! [M: list_P5561950507774946575_c_nat,K2: c,V4: nat] :
      ( ( distinct_c @ ( map_Pr72372295602292475_nat_c @ product_fst_c_nat @ M ) )
     => ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ K2 @ V4 ) @ ( set_Pr3395872032415823530_c_nat @ M ) )
       => ( ( mmap_lookup_c_nat @ M @ K2 )
          = ( some_nat @ V4 ) ) ) ) ).

% mmap_lookup_distinct
thf(fact_689_mmap__lookup__distinct,axiom,
    ! [M: list_P7417839048565863355_c_nat,K2: b,V4: produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ M ) )
     => ( ( member1522478204950355394_c_nat @ ( produc8868828497325608613_c_nat @ K2 @ V4 ) @ ( set_Pr5807476769692288598_c_nat @ M ) )
       => ( ( mmap_l5026999719965937591_c_nat @ M @ K2 )
          = ( some_P2720002978653898840_c_nat @ V4 ) ) ) ) ).

% mmap_lookup_distinct
thf(fact_690_mmap__lookup__distinct,axiom,
    ! [M: list_P903359562653991662od_b_c,K2: b,V4: c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ M ) )
     => ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ K2 @ V4 ) @ ( set_Product_prod_b_c2 @ M ) )
       => ( ( mmap_lookup_b_c @ M @ K2 )
          = ( some_c @ V4 ) ) ) ) ).

% mmap_lookup_distinct
thf(fact_691_mmap__lookup__distinct,axiom,
    ! [M: list_P8155366108056616461_c_nat,K2: b,V4: option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ M ) )
     => ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ V4 ) @ ( set_Pr5735414202003984296_c_nat @ M ) )
       => ( ( mmap_l1578203946870656521_c_nat @ M @ K2 )
          = ( some_o7552056839342673834_c_nat @ V4 ) ) ) ) ).

% mmap_lookup_distinct
thf(fact_692_eq__key__imp__eq__value,axiom,
    ! [Xs: list_P5561950507774946575_c_nat,K2: c,V1: nat,V22: nat] :
      ( ( distinct_c @ ( map_Pr72372295602292475_nat_c @ product_fst_c_nat @ Xs ) )
     => ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ K2 @ V1 ) @ ( set_Pr3395872032415823530_c_nat @ Xs ) )
       => ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ K2 @ V22 ) @ ( set_Pr3395872032415823530_c_nat @ Xs ) )
         => ( V1 = V22 ) ) ) ) ).

% eq_key_imp_eq_value
thf(fact_693_eq__key__imp__eq__value,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,K2: b,V1: produc4862256710654508797_c_nat,V22: produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs ) )
     => ( ( member1522478204950355394_c_nat @ ( produc8868828497325608613_c_nat @ K2 @ V1 ) @ ( set_Pr5807476769692288598_c_nat @ Xs ) )
       => ( ( member1522478204950355394_c_nat @ ( produc8868828497325608613_c_nat @ K2 @ V22 ) @ ( set_Pr5807476769692288598_c_nat @ Xs ) )
         => ( V1 = V22 ) ) ) ) ).

% eq_key_imp_eq_value
thf(fact_694_eq__key__imp__eq__value,axiom,
    ! [Xs: list_P903359562653991662od_b_c,K2: b,V1: c,V22: c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs ) )
     => ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ K2 @ V1 ) @ ( set_Product_prod_b_c2 @ Xs ) )
       => ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ K2 @ V22 ) @ ( set_Product_prod_b_c2 @ Xs ) )
         => ( V1 = V22 ) ) ) ) ).

% eq_key_imp_eq_value
thf(fact_695_eq__key__imp__eq__value,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,K2: b,V1: option7520157102916957007_c_nat,V22: option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs ) )
     => ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ V1 ) @ ( set_Pr5735414202003984296_c_nat @ Xs ) )
       => ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ V22 ) @ ( set_Pr5735414202003984296_c_nat @ Xs ) )
         => ( V1 = V22 ) ) ) ) ).

% eq_key_imp_eq_value
thf(fact_696_lookup__tabulate,axiom,
    ! [Xs: list_b,X4: b,F: b > produc4862256710654508797_c_nat] :
      ( ( distinct_b @ Xs )
     => ( ( ( member_b @ X4 @ ( set_b2 @ Xs ) )
         => ( ( lookup2261023941242044608_c_nat @ ( tabula8673640023562560852_c_nat @ Xs @ F ) @ X4 )
            = ( some_P2720002978653898840_c_nat @ ( F @ X4 ) ) ) )
        & ( ~ ( member_b @ X4 @ ( set_b2 @ Xs ) )
         => ( ( lookup2261023941242044608_c_nat @ ( tabula8673640023562560852_c_nat @ Xs @ F ) @ X4 )
            = none_P8487441334512977628_c_nat ) ) ) ) ).

% lookup_tabulate
thf(fact_697_fold__sup__def,axiom,
    ( fold_sup_c_nat
    = ( ^ [M2: list_P5561950507774946575_c_nat,F3: c > c] :
          ( mmap_fold_c_nat @ M2
          @ ( produc1049061115736377381_c_nat
            @ ^ [X3: c] : ( product_Pair_c_nat @ ( F3 @ X3 ) ) )
          @ sup_sup_nat
          @ nil_Pr650041864559007481_c_nat ) ) ) ).

% fold_sup_def
thf(fact_698_set__update__with__aux,axiom,
    ! [Xs: list_P5561950507774946575_c_nat,V4: nat,K2: c,F: nat > nat] :
      ( ( distinct_c @ ( map_Pr72372295602292475_nat_c @ product_fst_c_nat @ Xs ) )
     => ( ( set_Pr3395872032415823530_c_nat @ ( update5219137139480407212_nat_c @ V4 @ K2 @ F @ Xs ) )
        = ( sup_su2428977078917480673_c_nat
          @ ( minus_1376595062388175246_c_nat @ ( set_Pr3395872032415823530_c_nat @ Xs )
            @ ( product_Sigma_c_nat @ ( insert_c @ K2 @ bot_bot_set_c )
              @ ^ [Uu: c] : top_top_set_nat ) )
          @ ( insert1302120044651762863_c_nat
            @ ( product_Pair_c_nat @ K2
              @ ( F
                @ ( case_option_nat_nat @ V4
                  @ ^ [V: nat] : V
                  @ ( map_of_c_nat @ Xs @ K2 ) ) ) )
            @ bot_bo1794802125927853641_c_nat ) ) ) ) ).

% set_update_with_aux
thf(fact_699_set__update__with__aux,axiom,
    ! [Xs: list_P903359562653991662od_b_c,V4: c,K2: b,F: c > c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs ) )
     => ( ( set_Product_prod_b_c2 @ ( update_with_aux_c_b @ V4 @ K2 @ F @ Xs ) )
        = ( sup_su2554677861087362972od_b_c
          @ ( minus_6323455999294890031od_b_c @ ( set_Product_prod_b_c2 @ Xs )
            @ ( product_Sigma_b_c @ ( insert_b @ K2 @ bot_bot_set_b )
              @ ^ [Uu: b] : top_top_set_c ) )
          @ ( insert1747480808674937912od_b_c
            @ ( product_Pair_b_c @ K2
              @ ( F
                @ ( case_option_c_c @ V4
                  @ ^ [V: c] : V
                  @ ( map_of_b_c @ Xs @ K2 ) ) ) )
            @ bot_bo2863795366942399540od_b_c ) ) ) ) ).

% set_update_with_aux
thf(fact_700_set__update__with__aux,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,V4: option7520157102916957007_c_nat,K2: b,F: option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs ) )
     => ( ( set_Pr5735414202003984296_c_nat @ ( update6007670785845294700_nat_b @ V4 @ K2 @ F @ Xs ) )
        = ( sup_su1518086237216066783_c_nat
          @ ( minus_7724933661418357004_c_nat @ ( set_Pr5735414202003984296_c_nat @ Xs )
            @ ( produc1892124643397453190_c_nat @ ( insert_b @ K2 @ bot_bot_set_b )
              @ ^ [Uu: b] : top_to4621355910429214389_c_nat ) )
          @ ( insert8034245793825879341_c_nat
            @ ( produc5716802255202478839_c_nat @ K2
              @ ( F
                @ ( case_o5165501333728037931_c_nat @ V4
                  @ ^ [V: option7520157102916957007_c_nat] : V
                  @ ( map_of2598500733885366699_c_nat @ Xs @ K2 ) ) ) )
            @ bot_bo7474904860610844231_c_nat ) ) ) ) ).

% set_update_with_aux
thf(fact_701_set__update__with__aux,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,V4: produc4862256710654508797_c_nat,K2: b,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs ) )
     => ( ( set_Pr5807476769692288598_c_nat @ ( update5508768958394535230_nat_b @ V4 @ K2 @ F @ Xs ) )
        = ( sup_su2972426202601250957_c_nat
          @ ( minus_1848903822360895418_c_nat @ ( set_Pr5807476769692288598_c_nat @ Xs )
            @ ( produc237220987571061300_c_nat @ ( insert_b @ K2 @ bot_bot_set_b )
              @ ^ [Uu: b] : top_to7683274595066931299_c_nat ) )
          @ ( insert5634363330804326107_c_nat
            @ ( produc8868828497325608613_c_nat @ K2
              @ ( F
                @ ( case_o1602653449097480839_c_nat @ V4
                  @ ^ [V: produc4862256710654508797_c_nat] : V
                  @ ( map_of556380330368225625_c_nat @ Xs @ K2 ) ) ) )
            @ bot_bo1267386999437214197_c_nat ) ) ) ) ).

% set_update_with_aux
thf(fact_702_list_Omap__disc__iff,axiom,
    ! [F: produc6689720525226317995_c_nat > b,A: list_P7417839048565863355_c_nat] :
      ( ( ( map_Pr6200325787298368846_nat_b @ F @ A )
        = nil_b )
      = ( A = nil_Pr7000167559216756261_c_nat ) ) ).

% list.map_disc_iff
thf(fact_703_list_Omap__disc__iff,axiom,
    ! [F: product_prod_b_c > b,A: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ F @ A )
        = nil_b )
      = ( A = nil_Product_prod_b_c ) ) ).

% list.map_disc_iff
thf(fact_704_Nil__is__map__conv,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat] :
      ( ( nil_b
        = ( map_Pr6200325787298368846_nat_b @ F @ Xs ) )
      = ( Xs = nil_Pr7000167559216756261_c_nat ) ) ).

% Nil_is_map_conv
thf(fact_705_Nil__is__map__conv,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( nil_b
        = ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% Nil_is_map_conv
thf(fact_706_map__is__Nil__conv,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat] :
      ( ( ( map_Pr6200325787298368846_nat_b @ F @ Xs )
        = nil_b )
      = ( Xs = nil_Pr7000167559216756261_c_nat ) ) ).

% map_is_Nil_conv
thf(fact_707_map__is__Nil__conv,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
        = nil_b )
      = ( Xs = nil_Product_prod_b_c ) ) ).

% map_is_Nil_conv
thf(fact_708_map__of__is__SomeI,axiom,
    ! [Xys: list_P5561950507774946575_c_nat,X4: c,Y4: nat] :
      ( ( distinct_c @ ( map_Pr72372295602292475_nat_c @ product_fst_c_nat @ Xys ) )
     => ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X4 @ Y4 ) @ ( set_Pr3395872032415823530_c_nat @ Xys ) )
       => ( ( map_of_c_nat @ Xys @ X4 )
          = ( some_nat @ Y4 ) ) ) ) ).

% map_of_is_SomeI
thf(fact_709_map__of__is__SomeI,axiom,
    ! [Xys: list_P7417839048565863355_c_nat,X4: b,Y4: produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xys ) )
     => ( ( member1522478204950355394_c_nat @ ( produc8868828497325608613_c_nat @ X4 @ Y4 ) @ ( set_Pr5807476769692288598_c_nat @ Xys ) )
       => ( ( map_of556380330368225625_c_nat @ Xys @ X4 )
          = ( some_P2720002978653898840_c_nat @ Y4 ) ) ) ) ).

% map_of_is_SomeI
thf(fact_710_map__of__is__SomeI,axiom,
    ! [Xys: list_P903359562653991662od_b_c,X4: b,Y4: c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xys ) )
     => ( ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X4 @ Y4 ) @ ( set_Product_prod_b_c2 @ Xys ) )
       => ( ( map_of_b_c @ Xys @ X4 )
          = ( some_c @ Y4 ) ) ) ) ).

% map_of_is_SomeI
thf(fact_711_map__of__is__SomeI,axiom,
    ! [Xys: list_P8155366108056616461_c_nat,X4: b,Y4: option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xys ) )
     => ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) @ ( set_Pr5735414202003984296_c_nat @ Xys ) )
       => ( ( map_of2598500733885366699_c_nat @ Xys @ X4 )
          = ( some_o7552056839342673834_c_nat @ Y4 ) ) ) ) ).

% map_of_is_SomeI
thf(fact_712_Some__eq__map__of__iff,axiom,
    ! [Xys: list_P5561950507774946575_c_nat,Y4: nat,X4: c] :
      ( ( distinct_c @ ( map_Pr72372295602292475_nat_c @ product_fst_c_nat @ Xys ) )
     => ( ( ( some_nat @ Y4 )
          = ( map_of_c_nat @ Xys @ X4 ) )
        = ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X4 @ Y4 ) @ ( set_Pr3395872032415823530_c_nat @ Xys ) ) ) ) ).

% Some_eq_map_of_iff
thf(fact_713_Some__eq__map__of__iff,axiom,
    ! [Xys: list_P7417839048565863355_c_nat,Y4: produc4862256710654508797_c_nat,X4: b] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xys ) )
     => ( ( ( some_P2720002978653898840_c_nat @ Y4 )
          = ( map_of556380330368225625_c_nat @ Xys @ X4 ) )
        = ( member1522478204950355394_c_nat @ ( produc8868828497325608613_c_nat @ X4 @ Y4 ) @ ( set_Pr5807476769692288598_c_nat @ Xys ) ) ) ) ).

% Some_eq_map_of_iff
thf(fact_714_Some__eq__map__of__iff,axiom,
    ! [Xys: list_P903359562653991662od_b_c,Y4: c,X4: b] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xys ) )
     => ( ( ( some_c @ Y4 )
          = ( map_of_b_c @ Xys @ X4 ) )
        = ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X4 @ Y4 ) @ ( set_Product_prod_b_c2 @ Xys ) ) ) ) ).

% Some_eq_map_of_iff
thf(fact_715_Some__eq__map__of__iff,axiom,
    ! [Xys: list_P8155366108056616461_c_nat,Y4: option7520157102916957007_c_nat,X4: b] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xys ) )
     => ( ( ( some_o7552056839342673834_c_nat @ Y4 )
          = ( map_of2598500733885366699_c_nat @ Xys @ X4 ) )
        = ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) @ ( set_Pr5735414202003984296_c_nat @ Xys ) ) ) ) ).

% Some_eq_map_of_iff
thf(fact_716_map__of__eq__Some__iff,axiom,
    ! [Xys: list_P5561950507774946575_c_nat,X4: c,Y4: nat] :
      ( ( distinct_c @ ( map_Pr72372295602292475_nat_c @ product_fst_c_nat @ Xys ) )
     => ( ( ( map_of_c_nat @ Xys @ X4 )
          = ( some_nat @ Y4 ) )
        = ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X4 @ Y4 ) @ ( set_Pr3395872032415823530_c_nat @ Xys ) ) ) ) ).

% map_of_eq_Some_iff
thf(fact_717_map__of__eq__Some__iff,axiom,
    ! [Xys: list_P7417839048565863355_c_nat,X4: b,Y4: produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xys ) )
     => ( ( ( map_of556380330368225625_c_nat @ Xys @ X4 )
          = ( some_P2720002978653898840_c_nat @ Y4 ) )
        = ( member1522478204950355394_c_nat @ ( produc8868828497325608613_c_nat @ X4 @ Y4 ) @ ( set_Pr5807476769692288598_c_nat @ Xys ) ) ) ) ).

% map_of_eq_Some_iff
thf(fact_718_map__of__eq__Some__iff,axiom,
    ! [Xys: list_P903359562653991662od_b_c,X4: b,Y4: c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xys ) )
     => ( ( ( map_of_b_c @ Xys @ X4 )
          = ( some_c @ Y4 ) )
        = ( member7862447941013992593od_b_c @ ( product_Pair_b_c @ X4 @ Y4 ) @ ( set_Product_prod_b_c2 @ Xys ) ) ) ) ).

% map_of_eq_Some_iff
thf(fact_719_map__of__eq__Some__iff,axiom,
    ! [Xys: list_P8155366108056616461_c_nat,X4: b,Y4: option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xys ) )
     => ( ( ( map_of2598500733885366699_c_nat @ Xys @ X4 )
          = ( some_o7552056839342673834_c_nat @ Y4 ) )
        = ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) @ ( set_Pr5735414202003984296_c_nat @ Xys ) ) ) ) ).

% map_of_eq_Some_iff
thf(fact_720_set__diff__eq,axiom,
    ( minus_minus_set_b
    = ( ^ [A6: set_b,B6: set_b] :
          ( collect_b
          @ ^ [X3: b] :
              ( ( member_b @ X3 @ A6 )
              & ~ ( member_b @ X3 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_721_list_Osimps_I8_J,axiom,
    ! [F: produc6689720525226317995_c_nat > b] :
      ( ( map_Pr6200325787298368846_nat_b @ F @ nil_Pr7000167559216756261_c_nat )
      = nil_b ) ).

% list.simps(8)
thf(fact_722_list_Osimps_I8_J,axiom,
    ! [F: product_prod_b_c > b] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ nil_Product_prod_b_c )
      = nil_b ) ).

% list.simps(8)
thf(fact_723_mmap__lookup__def,axiom,
    mmap_l5026999719965937591_c_nat = map_of556380330368225625_c_nat ).

% mmap_lookup_def
thf(fact_724_weak__map__of__SomeI,axiom,
    ! [K2: b,X4: option7520157102916957007_c_nat,L2: list_P8155366108056616461_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ X4 ) @ ( set_Pr5735414202003984296_c_nat @ L2 ) )
     => ? [X: option7520157102916957007_c_nat] :
          ( ( map_of2598500733885366699_c_nat @ L2 @ K2 )
          = ( some_o7552056839342673834_c_nat @ X ) ) ) ).

% weak_map_of_SomeI
thf(fact_725_weak__map__of__SomeI,axiom,
    ! [K2: c,X4: nat,L2: list_P5561950507774946575_c_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ K2 @ X4 ) @ ( set_Pr3395872032415823530_c_nat @ L2 ) )
     => ? [X: nat] :
          ( ( map_of_c_nat @ L2 @ K2 )
          = ( some_nat @ X ) ) ) ).

% weak_map_of_SomeI
thf(fact_726_map__of__SomeD,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,K2: b,Y4: option7520157102916957007_c_nat] :
      ( ( ( map_of2598500733885366699_c_nat @ Xs @ K2 )
        = ( some_o7552056839342673834_c_nat @ Y4 ) )
     => ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ Y4 ) @ ( set_Pr5735414202003984296_c_nat @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_727_map__of__SomeD,axiom,
    ! [Xs: list_P5561950507774946575_c_nat,K2: c,Y4: nat] :
      ( ( ( map_of_c_nat @ Xs @ K2 )
        = ( some_nat @ Y4 ) )
     => ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ K2 @ Y4 ) @ ( set_Pr3395872032415823530_c_nat @ Xs ) ) ) ).

% map_of_SomeD
thf(fact_728_map__of__eq__dom,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,Ys: list_P7417839048565863355_c_nat] :
      ( ( ( map_of556380330368225625_c_nat @ Xs )
        = ( map_of556380330368225625_c_nat @ Ys ) )
     => ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Xs ) )
        = ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Ys ) ) ) ) ).

% map_of_eq_dom
thf(fact_729_map__of__eq__dom,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_of_b_c @ Xs )
        = ( map_of_b_c @ Ys ) )
     => ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Xs ) )
        = ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Ys ) ) ) ) ).

% map_of_eq_dom
thf(fact_730_map__of__eq__dom,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,Ys: list_P8155366108056616461_c_nat] :
      ( ( ( map_of2598500733885366699_c_nat @ Xs )
        = ( map_of2598500733885366699_c_nat @ Ys ) )
     => ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Xs ) )
        = ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Ys ) ) ) ) ).

% map_of_eq_dom
thf(fact_731_map__of__eqI,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,Ys: list_P7417839048565863355_c_nat] :
      ( ( ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs ) )
        = ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Ys ) ) )
     => ( ! [X: b] :
            ( ( member_b @ X @ ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs ) ) )
           => ( ( map_of556380330368225625_c_nat @ Xs @ X )
              = ( map_of556380330368225625_c_nat @ Ys @ X ) ) )
       => ( ( map_of556380330368225625_c_nat @ Xs )
          = ( map_of556380330368225625_c_nat @ Ys ) ) ) ) ).

% map_of_eqI
thf(fact_732_map__of__eqI,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs ) )
        = ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Ys ) ) )
     => ( ! [X: b] :
            ( ( member_b @ X @ ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs ) ) )
           => ( ( map_of_b_c @ Xs @ X )
              = ( map_of_b_c @ Ys @ X ) ) )
       => ( ( map_of_b_c @ Xs )
          = ( map_of_b_c @ Ys ) ) ) ) ).

% map_of_eqI
thf(fact_733_map__of__eqI,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,Ys: list_P8155366108056616461_c_nat] :
      ( ( ( set_b2 @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs ) )
        = ( set_b2 @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Ys ) ) )
     => ( ! [X: b] :
            ( ( member_b @ X @ ( set_b2 @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs ) ) )
           => ( ( map_of2598500733885366699_c_nat @ Xs @ X )
              = ( map_of2598500733885366699_c_nat @ Ys @ X ) ) )
       => ( ( map_of2598500733885366699_c_nat @ Xs )
          = ( map_of2598500733885366699_c_nat @ Ys ) ) ) ) ).

% map_of_eqI
thf(fact_734_map__of__inject__set,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,Ys: list_P7417839048565863355_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs ) )
     => ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Ys ) )
       => ( ( ( map_of556380330368225625_c_nat @ Xs )
            = ( map_of556380330368225625_c_nat @ Ys ) )
          = ( ( set_Pr5807476769692288598_c_nat @ Xs )
            = ( set_Pr5807476769692288598_c_nat @ Ys ) ) ) ) ) ).

% map_of_inject_set
thf(fact_735_map__of__inject__set,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs ) )
     => ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Ys ) )
       => ( ( ( map_of_b_c @ Xs )
            = ( map_of_b_c @ Ys ) )
          = ( ( set_Product_prod_b_c2 @ Xs )
            = ( set_Product_prod_b_c2 @ Ys ) ) ) ) ) ).

% map_of_inject_set
thf(fact_736_map__of__inject__set,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,Ys: list_P8155366108056616461_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs ) )
     => ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Ys ) )
       => ( ( ( map_of2598500733885366699_c_nat @ Xs )
            = ( map_of2598500733885366699_c_nat @ Ys ) )
          = ( ( set_Pr5735414202003984296_c_nat @ Xs )
            = ( set_Pr5735414202003984296_c_nat @ Ys ) ) ) ) ) ).

% map_of_inject_set
thf(fact_737_map__of__eq__None__iff,axiom,
    ! [Xys: list_P7417839048565863355_c_nat,X4: b] :
      ( ( ( map_of556380330368225625_c_nat @ Xys @ X4 )
        = none_P8487441334512977628_c_nat )
      = ( ~ ( member_b @ X4 @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Xys ) ) ) ) ) ).

% map_of_eq_None_iff
thf(fact_738_map__of__eq__None__iff,axiom,
    ! [Xys: list_P903359562653991662od_b_c,X4: b] :
      ( ( ( map_of_b_c @ Xys @ X4 )
        = none_c )
      = ( ~ ( member_b @ X4 @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Xys ) ) ) ) ) ).

% map_of_eq_None_iff
thf(fact_739_map__of__eq__None__iff,axiom,
    ! [Xys: list_P8155366108056616461_c_nat,X4: b] :
      ( ( ( map_of2598500733885366699_c_nat @ Xys @ X4 )
        = none_o1329286346417566254_c_nat )
      = ( ~ ( member_b @ X4 @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Xys ) ) ) ) ) ).

% map_of_eq_None_iff
thf(fact_740_in__image__insert__iff,axiom,
    ! [B5: set_set_b,X4: b,A4: set_b] :
      ( ! [C4: set_b] :
          ( ( member_set_b @ C4 @ B5 )
         => ~ ( member_b @ X4 @ C4 ) )
     => ( ( member_set_b @ A4 @ ( image_set_b_set_b @ ( insert_b @ X4 ) @ B5 ) )
        = ( ( member_b @ X4 @ A4 )
          & ( member_set_b @ ( minus_minus_set_b @ A4 @ ( insert_b @ X4 @ bot_bot_set_b ) ) @ B5 ) ) ) ) ).

% in_image_insert_iff
thf(fact_741_mmap__lookup__empty,axiom,
    ! [Z: b] :
      ( ( mmap_l5026999719965937591_c_nat @ nil_Pr7000167559216756261_c_nat @ Z )
      = none_P8487441334512977628_c_nat ) ).

% mmap_lookup_empty
thf(fact_742_map__of__zip__map,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,F: produc6689720525226317995_c_nat > b] :
      ( ( map_of4023600558955943791_nat_b @ ( zip_Pr6015829939078101403_nat_b @ Xs @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) ) )
      = ( ^ [X3: produc6689720525226317995_c_nat] : ( if_option_b @ ( member1522478204950355394_c_nat @ X3 @ ( set_Pr5807476769692288598_c_nat @ Xs ) ) @ ( some_b @ ( F @ X3 ) ) @ none_b ) ) ) ).

% map_of_zip_map
thf(fact_743_map__of__zip__map,axiom,
    ! [Xs: list_P903359562653991662od_b_c,F: product_prod_b_c > b] :
      ( ( map_of1979227509055838842_b_c_b @ ( zip_Pr4210574531626869070_b_c_b @ Xs @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) ) )
      = ( ^ [X3: product_prod_b_c] : ( if_option_b @ ( member7862447941013992593od_b_c @ X3 @ ( set_Product_prod_b_c2 @ Xs ) ) @ ( some_b @ ( F @ X3 ) ) @ none_b ) ) ) ).

% map_of_zip_map
thf(fact_744_map__of__zip__map,axiom,
    ! [Xs: list_b,F: b > produc4862256710654508797_c_nat] :
      ( ( map_of556380330368225625_c_nat @ ( zip_b_1292320801234550661_c_nat @ Xs @ ( map_b_6442276761558007352_c_nat @ F @ Xs ) ) )
      = ( ^ [X3: b] : ( if_opt6485458775533684743_c_nat @ ( member_b @ X3 @ ( set_b2 @ Xs ) ) @ ( some_P2720002978653898840_c_nat @ ( F @ X3 ) ) @ none_P8487441334512977628_c_nat ) ) ) ).

% map_of_zip_map
thf(fact_745_set__map__of__compr,axiom,
    ! [Xs: list_P7417839048565863355_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs ) )
     => ( ( set_Pr5807476769692288598_c_nat @ Xs )
        = ( collec8595031719707015680_c_nat
          @ ( produc6484547981188837958_nat_o
            @ ^ [K: b,V: produc4862256710654508797_c_nat] :
                ( ( map_of556380330368225625_c_nat @ Xs @ K )
                = ( some_P2720002978653898840_c_nat @ V ) ) ) ) ) ) ).

% set_map_of_compr
thf(fact_746_set__map__of__compr,axiom,
    ! [Xs: list_P903359562653991662od_b_c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs ) )
     => ( ( set_Product_prod_b_c2 @ Xs )
        = ( collec548942224018234067od_b_c
          @ ( produc1873627588981547243_b_c_o
            @ ^ [K: b,V: c] :
                ( ( map_of_b_c @ Xs @ K )
                = ( some_c @ V ) ) ) ) ) ) ).

% set_map_of_compr
thf(fact_747_set__map__of__compr,axiom,
    ! [Xs: list_P8155366108056616461_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs ) )
     => ( ( set_Pr5735414202003984296_c_nat @ Xs )
        = ( collec4170641529751027026_c_nat
          @ ( produc2544960137906461044_nat_o
            @ ^ [K: b,V: option7520157102916957007_c_nat] :
                ( ( map_of2598500733885366699_c_nat @ Xs @ K )
                = ( some_o7552056839342673834_c_nat @ V ) ) ) ) ) ) ).

% set_map_of_compr
thf(fact_748_mmap__fold__distinct,axiom,
    ! [M: list_P7417839048565863355_c_nat,F: produc6689720525226317995_c_nat > produc6689720525226317995_c_nat,C: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ M ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( mmap_f2405537622796709232_c_nat @ M @ F @ C @ nil_Pr7000167559216756261_c_nat ) ) ) ) ).

% mmap_fold_distinct
thf(fact_749_mmap__fold__distinct,axiom,
    ! [M: list_P903359562653991662od_b_c,F: product_prod_b_c > product_prod_b_c,C: c > c > c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ M ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( mmap_fold_b_c @ M @ F @ C @ nil_Product_prod_b_c ) ) ) ) ).

% mmap_fold_distinct
thf(fact_750_mmap__fold__distinct,axiom,
    ! [M: list_P8155366108056616461_c_nat,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,C: option7520157102916957007_c_nat > option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ M ) )
     => ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( mmap_f908472374020558018_c_nat @ M @ F @ C @ nil_Pr2407244426195543671_c_nat ) ) ) ) ).

% mmap_fold_distinct
thf(fact_751_set__delete__aux,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,K2: b] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs ) )
     => ( ( set_Pr5807476769692288598_c_nat @ ( delete9177780190517604182_c_nat @ K2 @ Xs ) )
        = ( minus_1848903822360895418_c_nat @ ( set_Pr5807476769692288598_c_nat @ Xs )
          @ ( produc237220987571061300_c_nat @ ( insert_b @ K2 @ bot_bot_set_b )
            @ ^ [Uu: b] : top_to7683274595066931299_c_nat ) ) ) ) ).

% set_delete_aux
thf(fact_752_set__delete__aux,axiom,
    ! [Xs: list_P903359562653991662od_b_c,K2: b] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs ) )
     => ( ( set_Product_prod_b_c2 @ ( delete_aux_b_c @ K2 @ Xs ) )
        = ( minus_6323455999294890031od_b_c @ ( set_Product_prod_b_c2 @ Xs )
          @ ( product_Sigma_b_c @ ( insert_b @ K2 @ bot_bot_set_b )
            @ ^ [Uu: b] : top_top_set_c ) ) ) ) ).

% set_delete_aux
thf(fact_753_set__delete__aux,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,K2: b] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs ) )
     => ( ( set_Pr5735414202003984296_c_nat @ ( delete7176321171147170984_c_nat @ K2 @ Xs ) )
        = ( minus_7724933661418357004_c_nat @ ( set_Pr5735414202003984296_c_nat @ Xs )
          @ ( produc1892124643397453190_c_nat @ ( insert_b @ K2 @ bot_bot_set_b )
            @ ^ [Uu: b] : top_to4621355910429214389_c_nat ) ) ) ) ).

% set_delete_aux
thf(fact_754_dom__delete__aux,axiom,
    ! [Ps: list_P7417839048565863355_c_nat,K2: b] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Ps ) )
     => ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ ( delete9177780190517604182_c_nat @ K2 @ Ps ) ) )
        = ( minus_minus_set_b @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Ps ) ) @ ( insert_b @ K2 @ bot_bot_set_b ) ) ) ) ).

% dom_delete_aux
thf(fact_755_dom__delete__aux,axiom,
    ! [Ps: list_P903359562653991662od_b_c,K2: b] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Ps ) )
     => ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( delete_aux_b_c @ K2 @ Ps ) ) )
        = ( minus_minus_set_b @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Ps ) ) @ ( insert_b @ K2 @ bot_bot_set_b ) ) ) ) ).

% dom_delete_aux
thf(fact_756_dom__delete__aux,axiom,
    ! [Ps: list_P8155366108056616461_c_nat,K2: b] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Ps ) )
     => ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ ( delete7176321171147170984_c_nat @ K2 @ Ps ) ) )
        = ( minus_minus_set_b @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Ps ) ) @ ( insert_b @ K2 @ bot_bot_set_b ) ) ) ) ).

% dom_delete_aux
thf(fact_757_ran__distinct,axiom,
    ! [Al: list_P7417839048565863355_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) )
     => ( ( ran_b_7972405368055877709_c_nat @ ( map_of556380330368225625_c_nat @ Al ) )
        = ( image_665133137765360559_c_nat @ produc6319817161620345591_c_nat @ ( set_Pr5807476769692288598_c_nat @ Al ) ) ) ) ).

% ran_distinct
thf(fact_758_ran__distinct,axiom,
    ! [Al: list_P903359562653991662od_b_c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) )
     => ( ( ran_b_c @ ( map_of_b_c @ Al ) )
        = ( image_7762865866877615261_b_c_c @ product_snd_b_c @ ( set_Product_prod_b_c2 @ Al ) ) ) ) ).

% ran_distinct
thf(fact_759_ran__distinct,axiom,
    ! [Al: list_P8155366108056616461_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) )
     => ( ( ran_b_4864047414971847839_c_nat @ ( map_of2598500733885366699_c_nat @ Al ) )
        = ( image_5441275730643634515_c_nat @ produc2435110264752509257_c_nat @ ( set_Pr5735414202003984296_c_nat @ Al ) ) ) ) ).

% ran_distinct
thf(fact_760_minus__set__def,axiom,
    ( minus_minus_set_b
    = ( ^ [A6: set_b,B6: set_b] :
          ( collect_b
          @ ( minus_minus_b_o
            @ ^ [X3: b] : ( member_b @ X3 @ A6 )
            @ ^ [X3: b] : ( member_b @ X3 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_761_distinct__delete__aux,axiom,
    ! [Ps: list_P7417839048565863355_c_nat,K2: b] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Ps ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( delete9177780190517604182_c_nat @ K2 @ Ps ) ) ) ) ).

% distinct_delete_aux
thf(fact_762_distinct__delete__aux,axiom,
    ! [Ps: list_P903359562653991662od_b_c,K2: b] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Ps ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( delete_aux_b_c @ K2 @ Ps ) ) ) ) ).

% distinct_delete_aux
thf(fact_763_distinct__delete__aux,axiom,
    ! [Ps: list_P8155366108056616461_c_nat,K2: b] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Ps ) )
     => ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( delete7176321171147170984_c_nat @ K2 @ Ps ) ) ) ) ).

% distinct_delete_aux
thf(fact_764_mmap__fold__set,axiom,
    ! [M: list_P7417839048565863355_c_nat,F: produc6689720525226317995_c_nat > produc6689720525226317995_c_nat,C: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ M ) )
     => ( ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( mmap_f2405537622796709232_c_nat @ M @ F @ C @ nil_Pr7000167559216756261_c_nat ) ) )
        = ( image_9118790927800078285_nat_b @ ( comp_P1437073467185880348_c_nat @ produc6033490199168946105_c_nat @ F ) @ ( set_Pr5807476769692288598_c_nat @ M ) ) ) ) ).

% mmap_fold_set
thf(fact_765_mmap__fold__set,axiom,
    ! [M: list_P903359562653991662od_b_c,F: product_prod_b_c > product_prod_b_c,C: c > c > c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ M ) )
     => ( ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( mmap_fold_b_c @ M @ F @ C @ nil_Product_prod_b_c ) ) )
        = ( image_7762865866877615260_b_c_b @ ( comp_P7908477705713330670od_b_c @ product_fst_b_c @ F ) @ ( set_Product_prod_b_c2 @ M ) ) ) ) ).

% mmap_fold_set
thf(fact_766_mmap__fold__set,axiom,
    ! [M: list_P8155366108056616461_c_nat,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,C: option7520157102916957007_c_nat > option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ M ) )
     => ( ( set_b2 @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( mmap_f908472374020558018_c_nat @ M @ F @ C @ nil_Pr2407244426195543671_c_nat ) ) )
        = ( image_2722527542522727419_nat_b @ ( comp_P4601747412748468892_c_nat @ produc8816143915943518219_c_nat @ F ) @ ( set_Pr5735414202003984296_c_nat @ M ) ) ) ) ).

% mmap_fold_set
thf(fact_767_dom__map__default,axiom,
    ! [K2: b,V4: produc4862256710654508797_c_nat,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Xs: list_P7417839048565863355_c_nat] :
      ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ ( map_de8565690227408816292_c_nat @ K2 @ V4 @ F @ Xs ) ) )
      = ( insert_b @ K2 @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Xs ) ) ) ) ).

% dom_map_default
thf(fact_768_dom__map__default,axiom,
    ! [K2: b,V4: c,F: c > c,Xs: list_P903359562653991662od_b_c] :
      ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( map_default_b_c @ K2 @ V4 @ F @ Xs ) ) )
      = ( insert_b @ K2 @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Xs ) ) ) ) ).

% dom_map_default
thf(fact_769_dom__map__default,axiom,
    ! [K2: b,V4: option7520157102916957007_c_nat,F: option7520157102916957007_c_nat > option7520157102916957007_c_nat,Xs: list_P8155366108056616461_c_nat] :
      ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ ( map_de5305620854455122934_c_nat @ K2 @ V4 @ F @ Xs ) ) )
      = ( insert_b @ K2 @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Xs ) ) ) ) ).

% dom_map_default
thf(fact_770_product__code,axiom,
    ! [Xs: list_b,Ys: list_o9079336047566018143_c_nat] :
      ( ( produc8039488468093475068_c_nat @ ( set_b2 @ Xs ) @ ( set_op620999267866781434_c_nat @ Ys ) )
      = ( set_Pr5735414202003984296_c_nat
        @ ( concat2980991215343123196_c_nat
          @ ( map_b_1379542981946909640_c_nat
            @ ^ [X3: b] : ( map_op2222718326440564242_c_nat @ ( produc5716802255202478839_c_nat @ X3 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_771_product__code,axiom,
    ! [Xs: list_c,Ys: list_nat] :
      ( ( produc8502899029972955646_c_nat @ ( set_c2 @ Xs ) @ ( set_nat2 @ Ys ) )
      = ( set_Pr3395872032415823530_c_nat
        @ ( concat185498358245221374_c_nat
          @ ( map_c_3181020965785344203_c_nat
            @ ^ [X3: c] : ( map_na7150870762157906453_c_nat @ ( product_Pair_c_nat @ X3 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_772_map__comp__map,axiom,
    ! [F: b > b,G: produc6689720525226317995_c_nat > b] :
      ( ( comp_l780237831682568542_c_nat @ ( map_b_b @ F ) @ ( map_Pr6200325787298368846_nat_b @ G ) )
      = ( map_Pr6200325787298368846_nat_b @ ( comp_b7259836426336601678_c_nat @ F @ G ) ) ) ).

% map_comp_map
thf(fact_773_map__comp__map,axiom,
    ! [F: b > b,G: product_prod_b_c > b] :
      ( ( comp_l7504339039855960459od_b_c @ ( map_b_b @ F ) @ ( map_Pr8222292247188198875_b_c_b @ G ) )
      = ( map_Pr8222292247188198875_b_c_b @ ( comp_b1099499051205000453od_b_c @ F @ G ) ) ) ).

% map_comp_map
thf(fact_774_map__comp__map,axiom,
    ! [F: produc6689720525226317995_c_nat > b,G: produc6689720525226317995_c_nat > produc6689720525226317995_c_nat] :
      ( ( comp_l7560420351980344226_c_nat @ ( map_Pr6200325787298368846_nat_b @ F ) @ ( map_Pr7976378254695727132_c_nat @ G ) )
      = ( map_Pr6200325787298368846_nat_b @ ( comp_P1437073467185880348_c_nat @ F @ G ) ) ) ).

% map_comp_map
thf(fact_775_map__comp__map,axiom,
    ! [F: produc6689720525226317995_c_nat > b,G: product_prod_b_c > produc6689720525226317995_c_nat] :
      ( ( comp_l1718039793250830919od_b_c @ ( map_Pr6200325787298368846_nat_b @ F ) @ ( map_Pr6155953585549333711_c_nat @ G ) )
      = ( map_Pr8222292247188198875_b_c_b @ ( comp_P6547017101245297847od_b_c @ F @ G ) ) ) ).

% map_comp_map
thf(fact_776_map__comp__map,axiom,
    ! [F: product_prod_b_c > b,G: produc6689720525226317995_c_nat > product_prod_b_c] :
      ( ( comp_l7667434161729046581_c_nat @ ( map_Pr8222292247188198875_b_c_b @ F ) @ ( map_Pr5949920527225541047od_b_c @ G ) )
      = ( map_Pr6200325787298368846_nat_b @ ( comp_P8916405840549426341_c_nat @ F @ G ) ) ) ).

% map_comp_map
thf(fact_777_map__comp__map,axiom,
    ! [F: product_prod_b_c > b,G: product_prod_b_c > product_prod_b_c] :
      ( ( comp_l4907364657749181940od_b_c @ ( map_Pr8222292247188198875_b_c_b @ F ) @ ( map_Pr6384634858812015940od_b_c @ G ) )
      = ( map_Pr8222292247188198875_b_c_b @ ( comp_P7908477705713330670od_b_c @ F @ G ) ) ) ).

% map_comp_map
thf(fact_778_List_Omap_Ocomp,axiom,
    ! [F: b > b,G: produc6689720525226317995_c_nat > b] :
      ( ( comp_l780237831682568542_c_nat @ ( map_b_b @ F ) @ ( map_Pr6200325787298368846_nat_b @ G ) )
      = ( map_Pr6200325787298368846_nat_b @ ( comp_b7259836426336601678_c_nat @ F @ G ) ) ) ).

% List.map.comp
thf(fact_779_List_Omap_Ocomp,axiom,
    ! [F: b > b,G: product_prod_b_c > b] :
      ( ( comp_l7504339039855960459od_b_c @ ( map_b_b @ F ) @ ( map_Pr8222292247188198875_b_c_b @ G ) )
      = ( map_Pr8222292247188198875_b_c_b @ ( comp_b1099499051205000453od_b_c @ F @ G ) ) ) ).

% List.map.comp
thf(fact_780_List_Omap_Ocomp,axiom,
    ! [F: produc6689720525226317995_c_nat > b,G: produc6689720525226317995_c_nat > produc6689720525226317995_c_nat] :
      ( ( comp_l7560420351980344226_c_nat @ ( map_Pr6200325787298368846_nat_b @ F ) @ ( map_Pr7976378254695727132_c_nat @ G ) )
      = ( map_Pr6200325787298368846_nat_b @ ( comp_P1437073467185880348_c_nat @ F @ G ) ) ) ).

% List.map.comp
thf(fact_781_List_Omap_Ocomp,axiom,
    ! [F: produc6689720525226317995_c_nat > b,G: product_prod_b_c > produc6689720525226317995_c_nat] :
      ( ( comp_l1718039793250830919od_b_c @ ( map_Pr6200325787298368846_nat_b @ F ) @ ( map_Pr6155953585549333711_c_nat @ G ) )
      = ( map_Pr8222292247188198875_b_c_b @ ( comp_P6547017101245297847od_b_c @ F @ G ) ) ) ).

% List.map.comp
thf(fact_782_List_Omap_Ocomp,axiom,
    ! [F: product_prod_b_c > b,G: produc6689720525226317995_c_nat > product_prod_b_c] :
      ( ( comp_l7667434161729046581_c_nat @ ( map_Pr8222292247188198875_b_c_b @ F ) @ ( map_Pr5949920527225541047od_b_c @ G ) )
      = ( map_Pr6200325787298368846_nat_b @ ( comp_P8916405840549426341_c_nat @ F @ G ) ) ) ).

% List.map.comp
thf(fact_783_List_Omap_Ocomp,axiom,
    ! [F: product_prod_b_c > b,G: product_prod_b_c > product_prod_b_c] :
      ( ( comp_l4907364657749181940od_b_c @ ( map_Pr8222292247188198875_b_c_b @ F ) @ ( map_Pr6384634858812015940od_b_c @ G ) )
      = ( map_Pr8222292247188198875_b_c_b @ ( comp_P7908477705713330670od_b_c @ F @ G ) ) ) ).

% List.map.comp
thf(fact_784_list_Omap__comp,axiom,
    ! [G: b > b,F: produc6689720525226317995_c_nat > b,V4: list_P7417839048565863355_c_nat] :
      ( ( map_b_b @ G @ ( map_Pr6200325787298368846_nat_b @ F @ V4 ) )
      = ( map_Pr6200325787298368846_nat_b @ ( comp_b7259836426336601678_c_nat @ G @ F ) @ V4 ) ) ).

% list.map_comp
thf(fact_785_list_Omap__comp,axiom,
    ! [G: b > b,F: product_prod_b_c > b,V4: list_P903359562653991662od_b_c] :
      ( ( map_b_b @ G @ ( map_Pr8222292247188198875_b_c_b @ F @ V4 ) )
      = ( map_Pr8222292247188198875_b_c_b @ ( comp_b1099499051205000453od_b_c @ G @ F ) @ V4 ) ) ).

% list.map_comp
thf(fact_786_list_Omap__comp,axiom,
    ! [G: produc6689720525226317995_c_nat > b,F: produc6689720525226317995_c_nat > produc6689720525226317995_c_nat,V4: list_P7417839048565863355_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ G @ ( map_Pr7976378254695727132_c_nat @ F @ V4 ) )
      = ( map_Pr6200325787298368846_nat_b @ ( comp_P1437073467185880348_c_nat @ G @ F ) @ V4 ) ) ).

% list.map_comp
thf(fact_787_list_Omap__comp,axiom,
    ! [G: produc6689720525226317995_c_nat > b,F: product_prod_b_c > produc6689720525226317995_c_nat,V4: list_P903359562653991662od_b_c] :
      ( ( map_Pr6200325787298368846_nat_b @ G @ ( map_Pr6155953585549333711_c_nat @ F @ V4 ) )
      = ( map_Pr8222292247188198875_b_c_b @ ( comp_P6547017101245297847od_b_c @ G @ F ) @ V4 ) ) ).

% list.map_comp
thf(fact_788_list_Omap__comp,axiom,
    ! [G: product_prod_b_c > b,F: produc6689720525226317995_c_nat > product_prod_b_c,V4: list_P7417839048565863355_c_nat] :
      ( ( map_Pr8222292247188198875_b_c_b @ G @ ( map_Pr5949920527225541047od_b_c @ F @ V4 ) )
      = ( map_Pr6200325787298368846_nat_b @ ( comp_P8916405840549426341_c_nat @ G @ F ) @ V4 ) ) ).

% list.map_comp
thf(fact_789_list_Omap__comp,axiom,
    ! [G: product_prod_b_c > b,F: product_prod_b_c > product_prod_b_c,V4: list_P903359562653991662od_b_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ G @ ( map_Pr6384634858812015940od_b_c @ F @ V4 ) )
      = ( map_Pr8222292247188198875_b_c_b @ ( comp_P7908477705713330670od_b_c @ G @ F ) @ V4 ) ) ).

% list.map_comp
thf(fact_790_List_Omap_Ocompositionality,axiom,
    ! [F: b > b,G: produc6689720525226317995_c_nat > b,List: list_P7417839048565863355_c_nat] :
      ( ( map_b_b @ F @ ( map_Pr6200325787298368846_nat_b @ G @ List ) )
      = ( map_Pr6200325787298368846_nat_b @ ( comp_b7259836426336601678_c_nat @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_791_List_Omap_Ocompositionality,axiom,
    ! [F: b > b,G: product_prod_b_c > b,List: list_P903359562653991662od_b_c] :
      ( ( map_b_b @ F @ ( map_Pr8222292247188198875_b_c_b @ G @ List ) )
      = ( map_Pr8222292247188198875_b_c_b @ ( comp_b1099499051205000453od_b_c @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_792_List_Omap_Ocompositionality,axiom,
    ! [F: produc6689720525226317995_c_nat > b,G: produc6689720525226317995_c_nat > produc6689720525226317995_c_nat,List: list_P7417839048565863355_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ F @ ( map_Pr7976378254695727132_c_nat @ G @ List ) )
      = ( map_Pr6200325787298368846_nat_b @ ( comp_P1437073467185880348_c_nat @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_793_List_Omap_Ocompositionality,axiom,
    ! [F: produc6689720525226317995_c_nat > b,G: product_prod_b_c > produc6689720525226317995_c_nat,List: list_P903359562653991662od_b_c] :
      ( ( map_Pr6200325787298368846_nat_b @ F @ ( map_Pr6155953585549333711_c_nat @ G @ List ) )
      = ( map_Pr8222292247188198875_b_c_b @ ( comp_P6547017101245297847od_b_c @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_794_List_Omap_Ocompositionality,axiom,
    ! [F: product_prod_b_c > b,G: produc6689720525226317995_c_nat > product_prod_b_c,List: list_P7417839048565863355_c_nat] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( map_Pr5949920527225541047od_b_c @ G @ List ) )
      = ( map_Pr6200325787298368846_nat_b @ ( comp_P8916405840549426341_c_nat @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_795_List_Omap_Ocompositionality,axiom,
    ! [F: product_prod_b_c > b,G: product_prod_b_c > product_prod_b_c,List: list_P903359562653991662od_b_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( map_Pr6384634858812015940od_b_c @ G @ List ) )
      = ( map_Pr8222292247188198875_b_c_b @ ( comp_P7908477705713330670od_b_c @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_796_map__map,axiom,
    ! [F: b > b,G: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat] :
      ( ( map_b_b @ F @ ( map_Pr6200325787298368846_nat_b @ G @ Xs ) )
      = ( map_Pr6200325787298368846_nat_b @ ( comp_b7259836426336601678_c_nat @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_797_map__map,axiom,
    ! [F: b > b,G: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( map_b_b @ F @ ( map_Pr8222292247188198875_b_c_b @ G @ Xs ) )
      = ( map_Pr8222292247188198875_b_c_b @ ( comp_b1099499051205000453od_b_c @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_798_map__map,axiom,
    ! [F: produc6689720525226317995_c_nat > b,G: produc6689720525226317995_c_nat > produc6689720525226317995_c_nat,Xs: list_P7417839048565863355_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ F @ ( map_Pr7976378254695727132_c_nat @ G @ Xs ) )
      = ( map_Pr6200325787298368846_nat_b @ ( comp_P1437073467185880348_c_nat @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_799_map__map,axiom,
    ! [F: produc6689720525226317995_c_nat > b,G: product_prod_b_c > produc6689720525226317995_c_nat,Xs: list_P903359562653991662od_b_c] :
      ( ( map_Pr6200325787298368846_nat_b @ F @ ( map_Pr6155953585549333711_c_nat @ G @ Xs ) )
      = ( map_Pr8222292247188198875_b_c_b @ ( comp_P6547017101245297847od_b_c @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_800_map__map,axiom,
    ! [F: product_prod_b_c > b,G: produc6689720525226317995_c_nat > product_prod_b_c,Xs: list_P7417839048565863355_c_nat] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( map_Pr5949920527225541047od_b_c @ G @ Xs ) )
      = ( map_Pr6200325787298368846_nat_b @ ( comp_P8916405840549426341_c_nat @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_801_map__map,axiom,
    ! [F: product_prod_b_c > b,G: product_prod_b_c > product_prod_b_c,Xs: list_P903359562653991662od_b_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( map_Pr6384634858812015940od_b_c @ G @ Xs ) )
      = ( map_Pr8222292247188198875_b_c_b @ ( comp_P7908477705713330670od_b_c @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_802_case__map__option,axiom,
    ! [G: produc4862256710654508797_c_nat,H: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,X4: option2970150418924381261_c_nat] :
      ( ( case_o1602653449097480839_c_nat @ G @ H @ ( map_op4586031498963288711_c_nat @ F @ X4 ) )
      = ( case_o1602653449097480839_c_nat @ G @ ( comp_P8480856022943104092_c_nat @ H @ F ) @ X4 ) ) ).

% case_map_option
thf(fact_803_case__map__option,axiom,
    ! [G: produc4862256710654508797_c_nat,H: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,F: product_prod_c_nat > produc4862256710654508797_c_nat,X4: option7520157102916957007_c_nat] :
      ( ( case_o1602653449097480839_c_nat @ G @ H @ ( map_op3508148262958903561_c_nat @ F @ X4 ) )
      = ( case_o6498799718091186953_c_nat @ G @ ( comp_P3772955433569424734_c_nat @ H @ F ) @ X4 ) ) ).

% case_map_option
thf(fact_804_case__map__option,axiom,
    ! [G: produc4862256710654508797_c_nat,H: product_prod_c_nat > produc4862256710654508797_c_nat,F: produc4862256710654508797_c_nat > product_prod_c_nat,X4: option2970150418924381261_c_nat] :
      ( ( case_o6498799718091186953_c_nat @ G @ H @ ( map_op6545395769035905801_c_nat @ F @ X4 ) )
      = ( case_o1602653449097480839_c_nat @ G @ ( comp_P1418391719297794398_c_nat @ H @ F ) @ X4 ) ) ).

% case_map_option
thf(fact_805_case__map__option,axiom,
    ! [G: produc4862256710654508797_c_nat,H: product_prod_c_nat > produc4862256710654508797_c_nat,F: product_prod_c_nat > product_prod_c_nat,X4: option7520157102916957007_c_nat] :
      ( ( case_o6498799718091186953_c_nat @ G @ H @ ( map_op2222230784418989707_c_nat @ F @ X4 ) )
      = ( case_o6498799718091186953_c_nat @ G @ ( comp_P591601820810598240_c_nat @ H @ F ) @ X4 ) ) ).

% case_map_option
thf(fact_806_map__ran__simps_I2_J,axiom,
    ! [F: b > option7520157102916957007_c_nat > option7520157102916957007_c_nat,K2: b,V4: option7520157102916957007_c_nat,Ps: list_P8155366108056616461_c_nat] :
      ( ( map_ra1379142644618337576_c_nat @ F @ ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ V4 ) @ Ps ) )
      = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ ( F @ K2 @ V4 ) ) @ ( map_ra1379142644618337576_c_nat @ F @ Ps ) ) ) ).

% map_ran_simps(2)
thf(fact_807_map__ran__simps_I2_J,axiom,
    ! [F: c > nat > nat,K2: c,V4: nat,Ps: list_P5561950507774946575_c_nat] :
      ( ( map_ran_c_nat_nat @ F @ ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K2 @ V4 ) @ Ps ) )
      = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K2 @ ( F @ K2 @ V4 ) ) @ ( map_ran_c_nat_nat @ F @ Ps ) ) ) ).

% map_ran_simps(2)
thf(fact_808_zip__Cons__Cons,axiom,
    ! [X4: b,Xs: list_b,Y4: option7520157102916957007_c_nat,Ys: list_o9079336047566018143_c_nat] :
      ( ( zip_b_1039741285442616279_c_nat @ ( cons_b @ X4 @ Xs ) @ ( cons_o7152427081109263641_c_nat @ Y4 @ Ys ) )
      = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) @ ( zip_b_1039741285442616279_c_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_809_zip__Cons__Cons,axiom,
    ! [X4: c,Xs: list_c,Y4: nat,Ys: list_nat] :
      ( ( zip_c_nat @ ( cons_c @ X4 @ Xs ) @ ( cons_nat @ Y4 @ Ys ) )
      = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ X4 @ Y4 ) @ ( zip_c_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_810_fst__comp__map__prod,axiom,
    ! [F: b > b,G: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( comp_P1437073467185880348_c_nat @ produc6033490199168946105_c_nat @ ( produc1006641225684822322_c_nat @ F @ G ) )
      = ( comp_b7259836426336601678_c_nat @ F @ produc6033490199168946105_c_nat ) ) ).

% fst_comp_map_prod
thf(fact_811_fst__comp__map__prod,axiom,
    ! [F: b > b,G: c > produc4862256710654508797_c_nat] :
      ( ( comp_P6547017101245297847od_b_c @ produc6033490199168946105_c_nat @ ( produc2579347470806298311_c_nat @ F @ G ) )
      = ( comp_b1099499051205000453od_b_c @ F @ product_fst_b_c ) ) ).

% fst_comp_map_prod
thf(fact_812_fst__comp__map__prod,axiom,
    ! [F: b > b,G: option7520157102916957007_c_nat > produc4862256710654508797_c_nat] :
      ( ( comp_P9073800064271228014_c_nat @ produc6033490199168946105_c_nat @ ( produc4824211994881367428_c_nat @ F @ G ) )
      = ( comp_b181056476299970464_c_nat @ F @ produc8816143915943518219_c_nat ) ) ).

% fst_comp_map_prod
thf(fact_813_fst__comp__map__prod,axiom,
    ! [F: b > b,G: produc4862256710654508797_c_nat > c] :
      ( ( comp_P8916405840549426341_c_nat @ product_fst_b_c @ ( produc2181303955692932747_nat_c @ F @ G ) )
      = ( comp_b7259836426336601678_c_nat @ F @ produc6033490199168946105_c_nat ) ) ).

% fst_comp_map_prod
thf(fact_814_fst__comp__map__prod,axiom,
    ! [F: b > b,G: c > c] :
      ( ( comp_P7908477705713330670od_b_c @ product_fst_b_c @ ( produc5527967938102404790_b_c_c @ F @ G ) )
      = ( comp_b1099499051205000453od_b_c @ F @ product_fst_b_c ) ) ).

% fst_comp_map_prod
thf(fact_815_fst__comp__map__prod,axiom,
    ! [F: b > b,G: option7520157102916957007_c_nat > c] :
      ( ( comp_P6972646978455201015_c_nat @ product_fst_b_c @ ( produc3202252859625268153_nat_c @ F @ G ) )
      = ( comp_b181056476299970464_c_nat @ F @ produc8816143915943518219_c_nat ) ) ).

% fst_comp_map_prod
thf(fact_816_fst__comp__map__prod,axiom,
    ! [F: b > b,G: produc4862256710654508797_c_nat > option7520157102916957007_c_nat] :
      ( ( comp_P7163099704581481802_c_nat @ produc8816143915943518219_c_nat @ ( produc2309332835414039172_c_nat @ F @ G ) )
      = ( comp_b7259836426336601678_c_nat @ F @ produc6033490199168946105_c_nat ) ) ).

% fst_comp_map_prod
thf(fact_817_fst__comp__map__prod,axiom,
    ! [F: b > b,G: c > option7520157102916957007_c_nat] :
      ( ( comp_P8474455673179243273od_b_c @ produc8816143915943518219_c_nat @ ( produc2445590922777410841_c_nat @ F @ G ) )
      = ( comp_b1099499051205000453od_b_c @ F @ product_fst_b_c ) ) ).

% fst_comp_map_prod
thf(fact_818_fst__comp__map__prod,axiom,
    ! [F: b > b,G: option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( comp_P4601747412748468892_c_nat @ produc8816143915943518219_c_nat @ ( produc3763450287754571990_c_nat @ F @ G ) )
      = ( comp_b181056476299970464_c_nat @ F @ produc8816143915943518219_c_nat ) ) ).

% fst_comp_map_prod
thf(fact_819_concat__map__singleton,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat] :
      ( ( concat_b
        @ ( map_Pr229572808336229844list_b
          @ ^ [X3: produc6689720525226317995_c_nat] : ( cons_b @ ( F @ X3 ) @ nil_b )
          @ Xs ) )
      = ( map_Pr6200325787298368846_nat_b @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_820_concat__map__singleton,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( concat_b
        @ ( map_Pr6622659525328036833list_b
          @ ^ [X3: product_prod_b_c] : ( cons_b @ ( F @ X3 ) @ nil_b )
          @ Xs ) )
      = ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_821_zip__eq__ConsE,axiom,
    ! [Xs: list_b,Ys: list_o9079336047566018143_c_nat,Xy2: produc4862256710654508797_c_nat,Xys: list_P8155366108056616461_c_nat] :
      ( ( ( zip_b_1039741285442616279_c_nat @ Xs @ Ys )
        = ( cons_P7928565014262433479_c_nat @ Xy2 @ Xys ) )
     => ~ ! [X: b,Xs3: list_b] :
            ( ( Xs
              = ( cons_b @ X @ Xs3 ) )
           => ! [Y: option7520157102916957007_c_nat,Ys3: list_o9079336047566018143_c_nat] :
                ( ( Ys
                  = ( cons_o7152427081109263641_c_nat @ Y @ Ys3 ) )
               => ( ( Xy2
                    = ( produc5716802255202478839_c_nat @ X @ Y ) )
                 => ( Xys
                   != ( zip_b_1039741285442616279_c_nat @ Xs3 @ Ys3 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_822_zip__eq__ConsE,axiom,
    ! [Xs: list_c,Ys: list_nat,Xy2: product_prod_c_nat,Xys: list_P5561950507774946575_c_nat] :
      ( ( ( zip_c_nat @ Xs @ Ys )
        = ( cons_P7676055461599297353_c_nat @ Xy2 @ Xys ) )
     => ~ ! [X: c,Xs3: list_c] :
            ( ( Xs
              = ( cons_c @ X @ Xs3 ) )
           => ! [Y: nat,Ys3: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y @ Ys3 ) )
               => ( ( Xy2
                    = ( product_Pair_c_nat @ X @ Y ) )
                 => ( Xys
                   != ( zip_c_nat @ Xs3 @ Ys3 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_823_list_Osimps_I9_J,axiom,
    ! [F: produc6689720525226317995_c_nat > b,X21: produc6689720525226317995_c_nat,X222: list_P7417839048565863355_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ F @ ( cons_P7294201633519847029_c_nat @ X21 @ X222 ) )
      = ( cons_b @ ( F @ X21 ) @ ( map_Pr6200325787298368846_nat_b @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_824_list_Osimps_I9_J,axiom,
    ! [F: product_prod_b_c > b,X21: product_prod_b_c,X222: list_P903359562653991662od_b_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( cons_P4529483553340347422od_b_c @ X21 @ X222 ) )
      = ( cons_b @ ( F @ X21 ) @ ( map_Pr8222292247188198875_b_c_b @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_825_Cons__eq__map__D,axiom,
    ! [X4: b,Xs: list_b,F: produc6689720525226317995_c_nat > b,Ys: list_P7417839048565863355_c_nat] :
      ( ( ( cons_b @ X4 @ Xs )
        = ( map_Pr6200325787298368846_nat_b @ F @ Ys ) )
     => ? [Z3: produc6689720525226317995_c_nat,Zs2: list_P7417839048565863355_c_nat] :
          ( ( Ys
            = ( cons_P7294201633519847029_c_nat @ Z3 @ Zs2 ) )
          & ( X4
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_Pr6200325787298368846_nat_b @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_826_Cons__eq__map__D,axiom,
    ! [X4: b,Xs: list_b,F: product_prod_b_c > b,Ys: list_P903359562653991662od_b_c] :
      ( ( ( cons_b @ X4 @ Xs )
        = ( map_Pr8222292247188198875_b_c_b @ F @ Ys ) )
     => ? [Z3: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
          ( ( Ys
            = ( cons_P4529483553340347422od_b_c @ Z3 @ Zs2 ) )
          & ( X4
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_Pr8222292247188198875_b_c_b @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_827_map__eq__Cons__D,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,Y4: b,Ys: list_b] :
      ( ( ( map_Pr6200325787298368846_nat_b @ F @ Xs )
        = ( cons_b @ Y4 @ Ys ) )
     => ? [Z3: produc6689720525226317995_c_nat,Zs2: list_P7417839048565863355_c_nat] :
          ( ( Xs
            = ( cons_P7294201633519847029_c_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y4 )
          & ( ( map_Pr6200325787298368846_nat_b @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_828_map__eq__Cons__D,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,Y4: b,Ys: list_b] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
        = ( cons_b @ Y4 @ Ys ) )
     => ? [Z3: product_prod_b_c,Zs2: list_P903359562653991662od_b_c] :
          ( ( Xs
            = ( cons_P4529483553340347422od_b_c @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y4 )
          & ( ( map_Pr8222292247188198875_b_c_b @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_829_Cons__eq__map__conv,axiom,
    ! [X4: b,Xs: list_b,F: produc6689720525226317995_c_nat > b,Ys: list_P7417839048565863355_c_nat] :
      ( ( ( cons_b @ X4 @ Xs )
        = ( map_Pr6200325787298368846_nat_b @ F @ Ys ) )
      = ( ? [Z4: produc6689720525226317995_c_nat,Zs3: list_P7417839048565863355_c_nat] :
            ( ( Ys
              = ( cons_P7294201633519847029_c_nat @ Z4 @ Zs3 ) )
            & ( X4
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_Pr6200325787298368846_nat_b @ F @ Zs3 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_830_Cons__eq__map__conv,axiom,
    ! [X4: b,Xs: list_b,F: product_prod_b_c > b,Ys: list_P903359562653991662od_b_c] :
      ( ( ( cons_b @ X4 @ Xs )
        = ( map_Pr8222292247188198875_b_c_b @ F @ Ys ) )
      = ( ? [Z4: product_prod_b_c,Zs3: list_P903359562653991662od_b_c] :
            ( ( Ys
              = ( cons_P4529483553340347422od_b_c @ Z4 @ Zs3 ) )
            & ( X4
              = ( F @ Z4 ) )
            & ( Xs
              = ( map_Pr8222292247188198875_b_c_b @ F @ Zs3 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_831_map__eq__Cons__conv,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,Y4: b,Ys: list_b] :
      ( ( ( map_Pr6200325787298368846_nat_b @ F @ Xs )
        = ( cons_b @ Y4 @ Ys ) )
      = ( ? [Z4: produc6689720525226317995_c_nat,Zs3: list_P7417839048565863355_c_nat] :
            ( ( Xs
              = ( cons_P7294201633519847029_c_nat @ Z4 @ Zs3 ) )
            & ( ( F @ Z4 )
              = Y4 )
            & ( ( map_Pr6200325787298368846_nat_b @ F @ Zs3 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_832_map__eq__Cons__conv,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,Y4: b,Ys: list_b] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
        = ( cons_b @ Y4 @ Ys ) )
      = ( ? [Z4: product_prod_b_c,Zs3: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( cons_P4529483553340347422od_b_c @ Z4 @ Zs3 ) )
            & ( ( F @ Z4 )
              = Y4 )
            & ( ( map_Pr8222292247188198875_b_c_b @ F @ Zs3 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_833_map__concat,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_l7485416851377393739_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ F @ ( concat6211037432900861354_c_nat @ Xs ) )
      = ( concat_b @ ( map_li1180621075770707268list_b @ ( map_Pr6200325787298368846_nat_b @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_834_map__concat,axiom,
    ! [F: product_prod_b_c > b,Xs: list_l8907847357763382004od_b_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( concat3284542956341965481od_b_c @ Xs ) )
      = ( concat_b @ ( map_li1600214420876558683list_b @ ( map_Pr8222292247188198875_b_c_b @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_835_map__default_Osimps_I1_J,axiom,
    ! [K2: b,V4: option7520157102916957007_c_nat,F: option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( map_de5305620854455122934_c_nat @ K2 @ V4 @ F @ nil_Pr2407244426195543671_c_nat )
      = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ V4 ) @ nil_Pr2407244426195543671_c_nat ) ) ).

% map_default.simps(1)
thf(fact_836_map__default_Osimps_I1_J,axiom,
    ! [K2: c,V4: nat,F: nat > nat] :
      ( ( map_default_c_nat @ K2 @ V4 @ F @ nil_Pr650041864559007481_c_nat )
      = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K2 @ V4 ) @ nil_Pr650041864559007481_c_nat ) ) ).

% map_default.simps(1)
thf(fact_837_map__default_Osimps_I2_J,axiom,
    ! [P: product_prod_c_nat,K2: c,V4: nat,F: nat > nat,Ps: list_P5561950507774946575_c_nat] :
      ( ( ( ( product_fst_c_nat @ P )
          = K2 )
       => ( ( map_default_c_nat @ K2 @ V4 @ F @ ( cons_P7676055461599297353_c_nat @ P @ Ps ) )
          = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K2 @ ( F @ ( product_snd_c_nat @ P ) ) ) @ Ps ) ) )
      & ( ( ( product_fst_c_nat @ P )
         != K2 )
       => ( ( map_default_c_nat @ K2 @ V4 @ F @ ( cons_P7676055461599297353_c_nat @ P @ Ps ) )
          = ( cons_P7676055461599297353_c_nat @ P @ ( map_default_c_nat @ K2 @ V4 @ F @ Ps ) ) ) ) ) ).

% map_default.simps(2)
thf(fact_838_map__default_Osimps_I2_J,axiom,
    ! [P: produc6689720525226317995_c_nat,K2: b,V4: produc4862256710654508797_c_nat,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Ps: list_P7417839048565863355_c_nat] :
      ( ( ( ( produc6033490199168946105_c_nat @ P )
          = K2 )
       => ( ( map_de8565690227408816292_c_nat @ K2 @ V4 @ F @ ( cons_P7294201633519847029_c_nat @ P @ Ps ) )
          = ( cons_P7294201633519847029_c_nat @ ( produc8868828497325608613_c_nat @ K2 @ ( F @ ( produc6319817161620345591_c_nat @ P ) ) ) @ Ps ) ) )
      & ( ( ( produc6033490199168946105_c_nat @ P )
         != K2 )
       => ( ( map_de8565690227408816292_c_nat @ K2 @ V4 @ F @ ( cons_P7294201633519847029_c_nat @ P @ Ps ) )
          = ( cons_P7294201633519847029_c_nat @ P @ ( map_de8565690227408816292_c_nat @ K2 @ V4 @ F @ Ps ) ) ) ) ) ).

% map_default.simps(2)
thf(fact_839_map__default_Osimps_I2_J,axiom,
    ! [P: product_prod_b_c,K2: b,V4: c,F: c > c,Ps: list_P903359562653991662od_b_c] :
      ( ( ( ( product_fst_b_c @ P )
          = K2 )
       => ( ( map_default_b_c @ K2 @ V4 @ F @ ( cons_P4529483553340347422od_b_c @ P @ Ps ) )
          = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ K2 @ ( F @ ( product_snd_b_c @ P ) ) ) @ Ps ) ) )
      & ( ( ( product_fst_b_c @ P )
         != K2 )
       => ( ( map_default_b_c @ K2 @ V4 @ F @ ( cons_P4529483553340347422od_b_c @ P @ Ps ) )
          = ( cons_P4529483553340347422od_b_c @ P @ ( map_default_b_c @ K2 @ V4 @ F @ Ps ) ) ) ) ) ).

% map_default.simps(2)
thf(fact_840_map__default_Osimps_I2_J,axiom,
    ! [P: produc4862256710654508797_c_nat,K2: b,V4: option7520157102916957007_c_nat,F: option7520157102916957007_c_nat > option7520157102916957007_c_nat,Ps: list_P8155366108056616461_c_nat] :
      ( ( ( ( produc8816143915943518219_c_nat @ P )
          = K2 )
       => ( ( map_de5305620854455122934_c_nat @ K2 @ V4 @ F @ ( cons_P7928565014262433479_c_nat @ P @ Ps ) )
          = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ ( F @ ( produc2435110264752509257_c_nat @ P ) ) ) @ Ps ) ) )
      & ( ( ( produc8816143915943518219_c_nat @ P )
         != K2 )
       => ( ( map_de5305620854455122934_c_nat @ K2 @ V4 @ F @ ( cons_P7928565014262433479_c_nat @ P @ Ps ) )
          = ( cons_P7928565014262433479_c_nat @ P @ ( map_de5305620854455122934_c_nat @ K2 @ V4 @ F @ Ps ) ) ) ) ) ).

% map_default.simps(2)
thf(fact_841_delete__aux_Ocases,axiom,
    ! [X4: produc6358558248090350139_c_nat] :
      ( ! [K3: b] :
          ( X4
         != ( produc385356081875622325_c_nat @ K3 @ nil_Pr2407244426195543671_c_nat ) )
     => ~ ! [K3: b,K4: b,V3: option7520157102916957007_c_nat,Xs4: list_P8155366108056616461_c_nat] :
            ( X4
           != ( produc385356081875622325_c_nat @ K3 @ ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K4 @ V3 ) @ Xs4 ) ) ) ) ).

% delete_aux.cases
thf(fact_842_delete__aux_Ocases,axiom,
    ! [X4: produc1501903449257105982_c_nat] :
      ( ! [K3: c] :
          ( X4
         != ( produc2615319020942666552_c_nat @ K3 @ nil_Pr650041864559007481_c_nat ) )
     => ~ ! [K3: c,K4: c,V3: nat,Xs4: list_P5561950507774946575_c_nat] :
            ( X4
           != ( produc2615319020942666552_c_nat @ K3 @ ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K4 @ V3 ) @ Xs4 ) ) ) ) ).

% delete_aux.cases
thf(fact_843_case__prod__o__map__prod,axiom,
    ! [F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,G1: b > b,G22: option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( comp_P8480856022943104092_c_nat @ ( produc3722540595886809633_c_nat @ F ) @ ( produc3763450287754571990_c_nat @ G1 @ G22 ) )
      = ( produc3722540595886809633_c_nat
        @ ^ [L: b,R3: option7520157102916957007_c_nat] : ( F @ ( G1 @ L ) @ ( G22 @ R3 ) ) ) ) ).

% case_prod_o_map_prod
thf(fact_844_case__prod__o__map__prod,axiom,
    ! [F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,G1: c > b,G22: nat > option7520157102916957007_c_nat] :
      ( ( comp_P3772955433569424734_c_nat @ ( produc3722540595886809633_c_nat @ F ) @ ( produc8240166225308521750_c_nat @ G1 @ G22 ) )
      = ( produc3293230613241633187_c_nat
        @ ^ [L: c,R3: nat] : ( F @ ( G1 @ L ) @ ( G22 @ R3 ) ) ) ) ).

% case_prod_o_map_prod
thf(fact_845_case__prod__o__map__prod,axiom,
    ! [F: c > nat > produc4862256710654508797_c_nat,G1: b > c,G22: option7520157102916957007_c_nat > nat] :
      ( ( comp_P1418391719297794398_c_nat @ ( produc3293230613241633187_c_nat @ F ) @ ( produc1238995859230195800at_nat @ G1 @ G22 ) )
      = ( produc3722540595886809633_c_nat
        @ ^ [L: b,R3: option7520157102916957007_c_nat] : ( F @ ( G1 @ L ) @ ( G22 @ R3 ) ) ) ) ).

% case_prod_o_map_prod
thf(fact_846_case__prod__o__map__prod,axiom,
    ! [F: c > nat > produc4862256710654508797_c_nat,G1: c > c,G22: nat > nat] :
      ( ( comp_P591601820810598240_c_nat @ ( produc3293230613241633187_c_nat @ F ) @ ( produc3634155571782648600at_nat @ G1 @ G22 ) )
      = ( produc3293230613241633187_c_nat
        @ ^ [L: c,R3: nat] : ( F @ ( G1 @ L ) @ ( G22 @ R3 ) ) ) ) ).

% case_prod_o_map_prod
thf(fact_847_delete__aux_Osimps_I2_J,axiom,
    ! [K2: b,K5: b,V4: option7520157102916957007_c_nat,Xs: list_P8155366108056616461_c_nat] :
      ( ( ( K2 = K5 )
       => ( ( delete7176321171147170984_c_nat @ K2 @ ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K5 @ V4 ) @ Xs ) )
          = Xs ) )
      & ( ( K2 != K5 )
       => ( ( delete7176321171147170984_c_nat @ K2 @ ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K5 @ V4 ) @ Xs ) )
          = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K5 @ V4 ) @ ( delete7176321171147170984_c_nat @ K2 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_848_delete__aux_Osimps_I2_J,axiom,
    ! [K2: c,K5: c,V4: nat,Xs: list_P5561950507774946575_c_nat] :
      ( ( ( K2 = K5 )
       => ( ( delete_aux_c_nat @ K2 @ ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K5 @ V4 ) @ Xs ) )
          = Xs ) )
      & ( ( K2 != K5 )
       => ( ( delete_aux_c_nat @ K2 @ ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K5 @ V4 ) @ Xs ) )
          = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K5 @ V4 ) @ ( delete_aux_c_nat @ K2 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_849_map__default_Oelims,axiom,
    ! [X4: c,Xa: nat,Xb: nat > nat,Xc: list_P5561950507774946575_c_nat,Y4: list_P5561950507774946575_c_nat] :
      ( ( ( map_default_c_nat @ X4 @ Xa @ Xb @ Xc )
        = Y4 )
     => ( ( ( Xc = nil_Pr650041864559007481_c_nat )
         => ( Y4
           != ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ X4 @ Xa ) @ nil_Pr650041864559007481_c_nat ) ) )
       => ~ ! [P6: product_prod_c_nat,Ps2: list_P5561950507774946575_c_nat] :
              ( ( Xc
                = ( cons_P7676055461599297353_c_nat @ P6 @ Ps2 ) )
             => ~ ( ( ( ( product_fst_c_nat @ P6 )
                      = X4 )
                   => ( Y4
                      = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ X4 @ ( Xb @ ( product_snd_c_nat @ P6 ) ) ) @ Ps2 ) ) )
                  & ( ( ( product_fst_c_nat @ P6 )
                     != X4 )
                   => ( Y4
                      = ( cons_P7676055461599297353_c_nat @ P6 @ ( map_default_c_nat @ X4 @ Xa @ Xb @ Ps2 ) ) ) ) ) ) ) ) ).

% map_default.elims
thf(fact_850_map__default_Oelims,axiom,
    ! [X4: b,Xa: produc4862256710654508797_c_nat,Xb: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Xc: list_P7417839048565863355_c_nat,Y4: list_P7417839048565863355_c_nat] :
      ( ( ( map_de8565690227408816292_c_nat @ X4 @ Xa @ Xb @ Xc )
        = Y4 )
     => ( ( ( Xc = nil_Pr7000167559216756261_c_nat )
         => ( Y4
           != ( cons_P7294201633519847029_c_nat @ ( produc8868828497325608613_c_nat @ X4 @ Xa ) @ nil_Pr7000167559216756261_c_nat ) ) )
       => ~ ! [P6: produc6689720525226317995_c_nat,Ps2: list_P7417839048565863355_c_nat] :
              ( ( Xc
                = ( cons_P7294201633519847029_c_nat @ P6 @ Ps2 ) )
             => ~ ( ( ( ( produc6033490199168946105_c_nat @ P6 )
                      = X4 )
                   => ( Y4
                      = ( cons_P7294201633519847029_c_nat @ ( produc8868828497325608613_c_nat @ X4 @ ( Xb @ ( produc6319817161620345591_c_nat @ P6 ) ) ) @ Ps2 ) ) )
                  & ( ( ( produc6033490199168946105_c_nat @ P6 )
                     != X4 )
                   => ( Y4
                      = ( cons_P7294201633519847029_c_nat @ P6 @ ( map_de8565690227408816292_c_nat @ X4 @ Xa @ Xb @ Ps2 ) ) ) ) ) ) ) ) ).

% map_default.elims
thf(fact_851_map__default_Oelims,axiom,
    ! [X4: b,Xa: c,Xb: c > c,Xc: list_P903359562653991662od_b_c,Y4: list_P903359562653991662od_b_c] :
      ( ( ( map_default_b_c @ X4 @ Xa @ Xb @ Xc )
        = Y4 )
     => ( ( ( Xc = nil_Product_prod_b_c )
         => ( Y4
           != ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ Xa ) @ nil_Product_prod_b_c ) ) )
       => ~ ! [P6: product_prod_b_c,Ps2: list_P903359562653991662od_b_c] :
              ( ( Xc
                = ( cons_P4529483553340347422od_b_c @ P6 @ Ps2 ) )
             => ~ ( ( ( ( product_fst_b_c @ P6 )
                      = X4 )
                   => ( Y4
                      = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ ( Xb @ ( product_snd_b_c @ P6 ) ) ) @ Ps2 ) ) )
                  & ( ( ( product_fst_b_c @ P6 )
                     != X4 )
                   => ( Y4
                      = ( cons_P4529483553340347422od_b_c @ P6 @ ( map_default_b_c @ X4 @ Xa @ Xb @ Ps2 ) ) ) ) ) ) ) ) ).

% map_default.elims
thf(fact_852_map__default_Oelims,axiom,
    ! [X4: b,Xa: option7520157102916957007_c_nat,Xb: option7520157102916957007_c_nat > option7520157102916957007_c_nat,Xc: list_P8155366108056616461_c_nat,Y4: list_P8155366108056616461_c_nat] :
      ( ( ( map_de5305620854455122934_c_nat @ X4 @ Xa @ Xb @ Xc )
        = Y4 )
     => ( ( ( Xc = nil_Pr2407244426195543671_c_nat )
         => ( Y4
           != ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Xa ) @ nil_Pr2407244426195543671_c_nat ) ) )
       => ~ ! [P6: produc4862256710654508797_c_nat,Ps2: list_P8155366108056616461_c_nat] :
              ( ( Xc
                = ( cons_P7928565014262433479_c_nat @ P6 @ Ps2 ) )
             => ~ ( ( ( ( produc8816143915943518219_c_nat @ P6 )
                      = X4 )
                   => ( Y4
                      = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ ( Xb @ ( produc2435110264752509257_c_nat @ P6 ) ) ) @ Ps2 ) ) )
                  & ( ( ( produc8816143915943518219_c_nat @ P6 )
                     != X4 )
                   => ( Y4
                      = ( cons_P7928565014262433479_c_nat @ P6 @ ( map_de5305620854455122934_c_nat @ X4 @ Xa @ Xb @ Ps2 ) ) ) ) ) ) ) ) ).

% map_default.elims
thf(fact_853_map__of__Cons__code_I2_J,axiom,
    ! [L2: b,K2: b,V4: option7520157102916957007_c_nat,Ps: list_P8155366108056616461_c_nat] :
      ( ( ( L2 = K2 )
       => ( ( map_of2598500733885366699_c_nat @ ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ L2 @ V4 ) @ Ps ) @ K2 )
          = ( some_o7552056839342673834_c_nat @ V4 ) ) )
      & ( ( L2 != K2 )
       => ( ( map_of2598500733885366699_c_nat @ ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ L2 @ V4 ) @ Ps ) @ K2 )
          = ( map_of2598500733885366699_c_nat @ Ps @ K2 ) ) ) ) ).

% map_of_Cons_code(2)
thf(fact_854_map__of__Cons__code_I2_J,axiom,
    ! [L2: c,K2: c,V4: nat,Ps: list_P5561950507774946575_c_nat] :
      ( ( ( L2 = K2 )
       => ( ( map_of_c_nat @ ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ L2 @ V4 ) @ Ps ) @ K2 )
          = ( some_nat @ V4 ) ) )
      & ( ( L2 != K2 )
       => ( ( map_of_c_nat @ ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ L2 @ V4 ) @ Ps ) @ K2 )
          = ( map_of_c_nat @ Ps @ K2 ) ) ) ) ).

% map_of_Cons_code(2)
thf(fact_855_delete__aux__eq__Nil__conv,axiom,
    ! [K2: b,Ts2: list_P8155366108056616461_c_nat] :
      ( ( ( delete7176321171147170984_c_nat @ K2 @ Ts2 )
        = nil_Pr2407244426195543671_c_nat )
      = ( ( Ts2 = nil_Pr2407244426195543671_c_nat )
        | ? [V: option7520157102916957007_c_nat] :
            ( Ts2
            = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ V ) @ nil_Pr2407244426195543671_c_nat ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_856_delete__aux__eq__Nil__conv,axiom,
    ! [K2: c,Ts2: list_P5561950507774946575_c_nat] :
      ( ( ( delete_aux_c_nat @ K2 @ Ts2 )
        = nil_Pr650041864559007481_c_nat )
      = ( ( Ts2 = nil_Pr650041864559007481_c_nat )
        | ? [V: nat] :
            ( Ts2
            = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K2 @ V ) @ nil_Pr650041864559007481_c_nat ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_857_delete__aux_Oelims,axiom,
    ! [X4: b,Xa: list_P8155366108056616461_c_nat,Y4: list_P8155366108056616461_c_nat] :
      ( ( ( delete7176321171147170984_c_nat @ X4 @ Xa )
        = Y4 )
     => ( ( ( Xa = nil_Pr2407244426195543671_c_nat )
         => ( Y4 != nil_Pr2407244426195543671_c_nat ) )
       => ~ ! [K4: b,V3: option7520157102916957007_c_nat,Xs4: list_P8155366108056616461_c_nat] :
              ( ( Xa
                = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K4 @ V3 ) @ Xs4 ) )
             => ~ ( ( ( X4 = K4 )
                   => ( Y4 = Xs4 ) )
                  & ( ( X4 != K4 )
                   => ( Y4
                      = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K4 @ V3 ) @ ( delete7176321171147170984_c_nat @ X4 @ Xs4 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_858_delete__aux_Oelims,axiom,
    ! [X4: c,Xa: list_P5561950507774946575_c_nat,Y4: list_P5561950507774946575_c_nat] :
      ( ( ( delete_aux_c_nat @ X4 @ Xa )
        = Y4 )
     => ( ( ( Xa = nil_Pr650041864559007481_c_nat )
         => ( Y4 != nil_Pr650041864559007481_c_nat ) )
       => ~ ! [K4: c,V3: nat,Xs4: list_P5561950507774946575_c_nat] :
              ( ( Xa
                = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K4 @ V3 ) @ Xs4 ) )
             => ~ ( ( ( X4 = K4 )
                   => ( Y4 = Xs4 ) )
                  & ( ( X4 != K4 )
                   => ( Y4
                      = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K4 @ V3 ) @ ( delete_aux_c_nat @ X4 @ Xs4 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_859_mmap__combine_Osimps_I1_J,axiom,
    ! [K2: b,V4: option7520157102916957007_c_nat,C: option7520157102916957007_c_nat > option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( mmap_c6869753822046002938_c_nat @ K2 @ V4 @ C @ nil_Pr2407244426195543671_c_nat )
      = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ V4 ) @ nil_Pr2407244426195543671_c_nat ) ) ).

% mmap_combine.simps(1)
thf(fact_860_mmap__combine_Osimps_I1_J,axiom,
    ! [K2: c,V4: nat,C: nat > nat > nat] :
      ( ( mmap_combine_c_nat @ K2 @ V4 @ C @ nil_Pr650041864559007481_c_nat )
      = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K2 @ V4 ) @ nil_Pr650041864559007481_c_nat ) ) ).

% mmap_combine.simps(1)
thf(fact_861_update__with__aux_Osimps_I1_J,axiom,
    ! [V4: option7520157102916957007_c_nat,K2: b,F: option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( update6007670785845294700_nat_b @ V4 @ K2 @ F @ nil_Pr2407244426195543671_c_nat )
      = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ ( F @ V4 ) ) @ nil_Pr2407244426195543671_c_nat ) ) ).

% update_with_aux.simps(1)
thf(fact_862_update__with__aux_Osimps_I1_J,axiom,
    ! [V4: nat,K2: c,F: nat > nat] :
      ( ( update5219137139480407212_nat_c @ V4 @ K2 @ F @ nil_Pr650041864559007481_c_nat )
      = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K2 @ ( F @ V4 ) ) @ nil_Pr650041864559007481_c_nat ) ) ).

% update_with_aux.simps(1)
thf(fact_863_product__concat__map,axiom,
    ( produc5715903733481046217_c_nat
    = ( ^ [Xs2: list_b,Ys2: list_o9079336047566018143_c_nat] :
          ( concat2980991215343123196_c_nat
          @ ( map_b_1379542981946909640_c_nat
            @ ^ [X3: b] : ( map_op2222718326440564242_c_nat @ ( produc5716802255202478839_c_nat @ X3 ) @ Ys2 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_864_product__concat__map,axiom,
    ( product_c_nat
    = ( ^ [Xs2: list_c,Ys2: list_nat] :
          ( concat185498358245221374_c_nat
          @ ( map_c_3181020965785344203_c_nat
            @ ^ [X3: c] : ( map_na7150870762157906453_c_nat @ ( product_Pair_c_nat @ X3 ) @ Ys2 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_865_fst__snd__flip,axiom,
    ( produc3722150186641245227_nat_b
    = ( comp_P6705017239471409046_nat_b @ produc2435110264752509257_c_nat
      @ ( produc2411843245718984257_c_nat
        @ ^ [X3: option7520157102916957007_c_nat,Y3: b] : ( produc5716802255202478839_c_nat @ Y3 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_866_fst__snd__flip,axiom,
    ( product_fst_nat_c
    = ( comp_P680640439754360793_nat_c @ product_snd_c_nat
      @ ( produc8194310905338674051_c_nat
        @ ^ [X3: nat,Y3: c] : ( product_Pair_c_nat @ Y3 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_867_fst__snd__flip,axiom,
    ( produc6033490199168946105_c_nat
    = ( comp_P46617119559061008_c_nat @ produc296059414264809275_nat_b
      @ ( produc1820909744133759113_nat_b
        @ ^ [X3: b,Y3: produc4862256710654508797_c_nat] : ( produc2845070749970072297_nat_b @ Y3 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_868_fst__snd__flip,axiom,
    ( product_fst_b_c
    = ( comp_P6635008700364650412od_b_c @ product_snd_c_b
      @ ( produc6717796504295817721od_c_b
        @ ^ [X3: b,Y3: c] : ( product_Pair_c_b @ Y3 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_869_fst__snd__flip,axiom,
    ( produc8816143915943518219_c_nat
    = ( comp_P289832891244043188_c_nat @ produc6564488572305012073_nat_b
      @ ( produc6406145797830999817_nat_b
        @ ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( produc622808525900205847_nat_b @ Y3 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_870_snd__fst__flip,axiom,
    ( product_snd_nat_c
    = ( comp_P1391869959443764803_nat_c @ product_fst_c_nat
      @ ( produc8194310905338674051_c_nat
        @ ^ [X3: nat,Y3: c] : ( product_Pair_c_nat @ Y3 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_871_snd__fst__flip,axiom,
    ( produc296059414264809275_nat_b
    = ( comp_P7114124675592174888_nat_b @ produc6033490199168946105_c_nat
      @ ( produc5941409774317716161_c_nat
        @ ^ [X3: produc4862256710654508797_c_nat,Y3: b] : ( produc8868828497325608613_c_nat @ Y3 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_872_snd__fst__flip,axiom,
    ( product_snd_c_b
    = ( comp_P5121022119437425836od_c_b @ product_fst_b_c
      @ ( produc4861268940673599101od_b_c
        @ ^ [X3: c,Y3: b] : ( product_Pair_b_c @ Y3 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_873_snd__fst__flip,axiom,
    ( produc6564488572305012073_nat_b
    = ( comp_P7285352614692659076_nat_b @ produc8816143915943518219_c_nat
      @ ( produc2411843245718984257_c_nat
        @ ^ [X3: option7520157102916957007_c_nat,Y3: b] : ( produc5716802255202478839_c_nat @ Y3 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_874_map__ran__Cons__sel,axiom,
    ! [F: b > produc4862256710654508797_c_nat > option7520157102916957007_c_nat,P: produc6689720525226317995_c_nat,Ps: list_P7417839048565863355_c_nat] :
      ( ( map_ra3237904418225743062_c_nat @ F @ ( cons_P7294201633519847029_c_nat @ P @ Ps ) )
      = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( produc6033490199168946105_c_nat @ P ) @ ( F @ ( produc6033490199168946105_c_nat @ P ) @ ( produc6319817161620345591_c_nat @ P ) ) ) @ ( map_ra3237904418225743062_c_nat @ F @ Ps ) ) ) ).

% map_ran_Cons_sel
thf(fact_875_map__ran__Cons__sel,axiom,
    ! [F: b > c > option7520157102916957007_c_nat,P: product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
      ( ( map_ra2860756734228053063_c_nat @ F @ ( cons_P4529483553340347422od_b_c @ P @ Ps ) )
      = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( product_fst_b_c @ P ) @ ( F @ ( product_fst_b_c @ P ) @ ( product_snd_b_c @ P ) ) ) @ ( map_ra2860756734228053063_c_nat @ F @ Ps ) ) ) ).

% map_ran_Cons_sel
thf(fact_876_map__ran__Cons__sel,axiom,
    ! [F: b > option7520157102916957007_c_nat > option7520157102916957007_c_nat,P: produc4862256710654508797_c_nat,Ps: list_P8155366108056616461_c_nat] :
      ( ( map_ra1379142644618337576_c_nat @ F @ ( cons_P7928565014262433479_c_nat @ P @ Ps ) )
      = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( produc8816143915943518219_c_nat @ P ) @ ( F @ ( produc8816143915943518219_c_nat @ P ) @ ( produc2435110264752509257_c_nat @ P ) ) ) @ ( map_ra1379142644618337576_c_nat @ F @ Ps ) ) ) ).

% map_ran_Cons_sel
thf(fact_877_update__with__aux_Osimps_I2_J,axiom,
    ! [P: product_prod_c_nat,K2: c,V4: nat,F: nat > nat,Ps: list_P5561950507774946575_c_nat] :
      ( ( ( ( product_fst_c_nat @ P )
          = K2 )
       => ( ( update5219137139480407212_nat_c @ V4 @ K2 @ F @ ( cons_P7676055461599297353_c_nat @ P @ Ps ) )
          = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K2 @ ( F @ ( product_snd_c_nat @ P ) ) ) @ Ps ) ) )
      & ( ( ( product_fst_c_nat @ P )
         != K2 )
       => ( ( update5219137139480407212_nat_c @ V4 @ K2 @ F @ ( cons_P7676055461599297353_c_nat @ P @ Ps ) )
          = ( cons_P7676055461599297353_c_nat @ P @ ( update5219137139480407212_nat_c @ V4 @ K2 @ F @ Ps ) ) ) ) ) ).

% update_with_aux.simps(2)
thf(fact_878_update__with__aux_Osimps_I2_J,axiom,
    ! [P: produc6689720525226317995_c_nat,K2: b,V4: produc4862256710654508797_c_nat,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Ps: list_P7417839048565863355_c_nat] :
      ( ( ( ( produc6033490199168946105_c_nat @ P )
          = K2 )
       => ( ( update5508768958394535230_nat_b @ V4 @ K2 @ F @ ( cons_P7294201633519847029_c_nat @ P @ Ps ) )
          = ( cons_P7294201633519847029_c_nat @ ( produc8868828497325608613_c_nat @ K2 @ ( F @ ( produc6319817161620345591_c_nat @ P ) ) ) @ Ps ) ) )
      & ( ( ( produc6033490199168946105_c_nat @ P )
         != K2 )
       => ( ( update5508768958394535230_nat_b @ V4 @ K2 @ F @ ( cons_P7294201633519847029_c_nat @ P @ Ps ) )
          = ( cons_P7294201633519847029_c_nat @ P @ ( update5508768958394535230_nat_b @ V4 @ K2 @ F @ Ps ) ) ) ) ) ).

% update_with_aux.simps(2)
thf(fact_879_update__with__aux_Osimps_I2_J,axiom,
    ! [P: product_prod_b_c,K2: b,V4: c,F: c > c,Ps: list_P903359562653991662od_b_c] :
      ( ( ( ( product_fst_b_c @ P )
          = K2 )
       => ( ( update_with_aux_c_b @ V4 @ K2 @ F @ ( cons_P4529483553340347422od_b_c @ P @ Ps ) )
          = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ K2 @ ( F @ ( product_snd_b_c @ P ) ) ) @ Ps ) ) )
      & ( ( ( product_fst_b_c @ P )
         != K2 )
       => ( ( update_with_aux_c_b @ V4 @ K2 @ F @ ( cons_P4529483553340347422od_b_c @ P @ Ps ) )
          = ( cons_P4529483553340347422od_b_c @ P @ ( update_with_aux_c_b @ V4 @ K2 @ F @ Ps ) ) ) ) ) ).

% update_with_aux.simps(2)
thf(fact_880_update__with__aux_Osimps_I2_J,axiom,
    ! [P: produc4862256710654508797_c_nat,K2: b,V4: option7520157102916957007_c_nat,F: option7520157102916957007_c_nat > option7520157102916957007_c_nat,Ps: list_P8155366108056616461_c_nat] :
      ( ( ( ( produc8816143915943518219_c_nat @ P )
          = K2 )
       => ( ( update6007670785845294700_nat_b @ V4 @ K2 @ F @ ( cons_P7928565014262433479_c_nat @ P @ Ps ) )
          = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ ( F @ ( produc2435110264752509257_c_nat @ P ) ) ) @ Ps ) ) )
      & ( ( ( produc8816143915943518219_c_nat @ P )
         != K2 )
       => ( ( update6007670785845294700_nat_b @ V4 @ K2 @ F @ ( cons_P7928565014262433479_c_nat @ P @ Ps ) )
          = ( cons_P7928565014262433479_c_nat @ P @ ( update6007670785845294700_nat_b @ V4 @ K2 @ F @ Ps ) ) ) ) ) ).

% update_with_aux.simps(2)
thf(fact_881_mmap__combine_Osimps_I2_J,axiom,
    ! [K2: b,V4: option7520157102916957007_c_nat,C: option7520157102916957007_c_nat > option7520157102916957007_c_nat > option7520157102916957007_c_nat,P: produc4862256710654508797_c_nat,Ps: list_P8155366108056616461_c_nat] :
      ( ( mmap_c6869753822046002938_c_nat @ K2 @ V4 @ C @ ( cons_P7928565014262433479_c_nat @ P @ Ps ) )
      = ( produc1076207890078318385_c_nat
        @ ^ [K6: b,V5: option7520157102916957007_c_nat] : ( if_lis6091463167075048391_c_nat @ ( K2 = K6 ) @ ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ ( C @ V5 @ V4 ) ) @ Ps ) @ ( cons_P7928565014262433479_c_nat @ P @ ( mmap_c6869753822046002938_c_nat @ K2 @ V4 @ C @ Ps ) ) )
        @ P ) ) ).

% mmap_combine.simps(2)
thf(fact_882_mmap__combine_Osimps_I2_J,axiom,
    ! [K2: c,V4: nat,C: nat > nat > nat,P: product_prod_c_nat,Ps: list_P5561950507774946575_c_nat] :
      ( ( mmap_combine_c_nat @ K2 @ V4 @ C @ ( cons_P7676055461599297353_c_nat @ P @ Ps ) )
      = ( produc5254646174846288821_c_nat
        @ ^ [K6: c,V5: nat] : ( if_lis8000280747373556681_c_nat @ ( K2 = K6 ) @ ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K2 @ ( C @ V5 @ V4 ) ) @ Ps ) @ ( cons_P7676055461599297353_c_nat @ P @ ( mmap_combine_c_nat @ K2 @ V4 @ C @ Ps ) ) )
        @ P ) ) ).

% mmap_combine.simps(2)
thf(fact_883_distinct__map__default,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,K2: b,V4: produc4862256710654508797_c_nat,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( map_de8565690227408816292_c_nat @ K2 @ V4 @ F @ Xs ) ) ) ) ).

% distinct_map_default
thf(fact_884_distinct__map__default,axiom,
    ! [Xs: list_P903359562653991662od_b_c,K2: b,V4: c,F: c > c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( map_default_b_c @ K2 @ V4 @ F @ Xs ) ) ) ) ).

% distinct_map_default
thf(fact_885_distinct__map__default,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,K2: b,V4: option7520157102916957007_c_nat,F: option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs ) )
     => ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( map_de5305620854455122934_c_nat @ K2 @ V4 @ F @ Xs ) ) ) ) ).

% distinct_map_default
thf(fact_886_foldl__mmap__combine__distinct__set,axiom,
    ! [R2: list_P7417839048565863355_c_nat,M: list_P7417839048565863355_c_nat,F: produc6689720525226317995_c_nat > produc6689720525226317995_c_nat,C: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ R2 ) )
     => ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( mmap_f2405537622796709232_c_nat @ M @ F @ C @ R2 ) ) )
        & ( ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( mmap_f2405537622796709232_c_nat @ M @ F @ C @ R2 ) ) )
          = ( sup_sup_set_b @ ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ R2 ) ) @ ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ ( comp_P1437073467185880348_c_nat @ produc6033490199168946105_c_nat @ F ) @ M ) ) ) ) ) ) ).

% foldl_mmap_combine_distinct_set
thf(fact_887_foldl__mmap__combine__distinct__set,axiom,
    ! [R2: list_P903359562653991662od_b_c,M: list_P903359562653991662od_b_c,F: product_prod_b_c > product_prod_b_c,C: c > c > c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ R2 ) )
     => ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( mmap_fold_b_c @ M @ F @ C @ R2 ) ) )
        & ( ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( mmap_fold_b_c @ M @ F @ C @ R2 ) ) )
          = ( sup_sup_set_b @ ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ R2 ) ) @ ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ ( comp_P7908477705713330670od_b_c @ product_fst_b_c @ F ) @ M ) ) ) ) ) ) ).

% foldl_mmap_combine_distinct_set
thf(fact_888_foldl__mmap__combine__distinct__set,axiom,
    ! [R2: list_P8155366108056616461_c_nat,M: list_P8155366108056616461_c_nat,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,C: option7520157102916957007_c_nat > option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ R2 ) )
     => ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( mmap_f908472374020558018_c_nat @ M @ F @ C @ R2 ) ) )
        & ( ( set_b2 @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( mmap_f908472374020558018_c_nat @ M @ F @ C @ R2 ) ) )
          = ( sup_sup_set_b @ ( set_b2 @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ R2 ) ) @ ( set_b2 @ ( map_Pr418519014202471036_nat_b @ ( comp_P4601747412748468892_c_nat @ produc8816143915943518219_c_nat @ F ) @ M ) ) ) ) ) ) ).

% foldl_mmap_combine_distinct_set
thf(fact_889_map__entry_Oelims,axiom,
    ! [X4: c,Xa: nat > nat,Xb: list_P5561950507774946575_c_nat,Y4: list_P5561950507774946575_c_nat] :
      ( ( ( map_entry_c_nat @ X4 @ Xa @ Xb )
        = Y4 )
     => ( ( ( Xb = nil_Pr650041864559007481_c_nat )
         => ( Y4 != nil_Pr650041864559007481_c_nat ) )
       => ~ ! [P6: product_prod_c_nat,Ps2: list_P5561950507774946575_c_nat] :
              ( ( Xb
                = ( cons_P7676055461599297353_c_nat @ P6 @ Ps2 ) )
             => ~ ( ( ( ( product_fst_c_nat @ P6 )
                      = X4 )
                   => ( Y4
                      = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ X4 @ ( Xa @ ( product_snd_c_nat @ P6 ) ) ) @ Ps2 ) ) )
                  & ( ( ( product_fst_c_nat @ P6 )
                     != X4 )
                   => ( Y4
                      = ( cons_P7676055461599297353_c_nat @ P6 @ ( map_entry_c_nat @ X4 @ Xa @ Ps2 ) ) ) ) ) ) ) ) ).

% map_entry.elims
thf(fact_890_map__entry_Oelims,axiom,
    ! [X4: b,Xa: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Xb: list_P7417839048565863355_c_nat,Y4: list_P7417839048565863355_c_nat] :
      ( ( ( map_en8198299921509431923_c_nat @ X4 @ Xa @ Xb )
        = Y4 )
     => ( ( ( Xb = nil_Pr7000167559216756261_c_nat )
         => ( Y4 != nil_Pr7000167559216756261_c_nat ) )
       => ~ ! [P6: produc6689720525226317995_c_nat,Ps2: list_P7417839048565863355_c_nat] :
              ( ( Xb
                = ( cons_P7294201633519847029_c_nat @ P6 @ Ps2 ) )
             => ~ ( ( ( ( produc6033490199168946105_c_nat @ P6 )
                      = X4 )
                   => ( Y4
                      = ( cons_P7294201633519847029_c_nat @ ( produc8868828497325608613_c_nat @ X4 @ ( Xa @ ( produc6319817161620345591_c_nat @ P6 ) ) ) @ Ps2 ) ) )
                  & ( ( ( produc6033490199168946105_c_nat @ P6 )
                     != X4 )
                   => ( Y4
                      = ( cons_P7294201633519847029_c_nat @ P6 @ ( map_en8198299921509431923_c_nat @ X4 @ Xa @ Ps2 ) ) ) ) ) ) ) ) ).

% map_entry.elims
thf(fact_891_map__entry_Oelims,axiom,
    ! [X4: b,Xa: c > c,Xb: list_P903359562653991662od_b_c,Y4: list_P903359562653991662od_b_c] :
      ( ( ( map_entry_b_c @ X4 @ Xa @ Xb )
        = Y4 )
     => ( ( ( Xb = nil_Product_prod_b_c )
         => ( Y4 != nil_Product_prod_b_c ) )
       => ~ ! [P6: product_prod_b_c,Ps2: list_P903359562653991662od_b_c] :
              ( ( Xb
                = ( cons_P4529483553340347422od_b_c @ P6 @ Ps2 ) )
             => ~ ( ( ( ( product_fst_b_c @ P6 )
                      = X4 )
                   => ( Y4
                      = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ ( Xa @ ( product_snd_b_c @ P6 ) ) ) @ Ps2 ) ) )
                  & ( ( ( product_fst_b_c @ P6 )
                     != X4 )
                   => ( Y4
                      = ( cons_P4529483553340347422od_b_c @ P6 @ ( map_entry_b_c @ X4 @ Xa @ Ps2 ) ) ) ) ) ) ) ) ).

% map_entry.elims
thf(fact_892_map__entry_Oelims,axiom,
    ! [X4: b,Xa: option7520157102916957007_c_nat > option7520157102916957007_c_nat,Xb: list_P8155366108056616461_c_nat,Y4: list_P8155366108056616461_c_nat] :
      ( ( ( map_en7725529839661896901_c_nat @ X4 @ Xa @ Xb )
        = Y4 )
     => ( ( ( Xb = nil_Pr2407244426195543671_c_nat )
         => ( Y4 != nil_Pr2407244426195543671_c_nat ) )
       => ~ ! [P6: produc4862256710654508797_c_nat,Ps2: list_P8155366108056616461_c_nat] :
              ( ( Xb
                = ( cons_P7928565014262433479_c_nat @ P6 @ Ps2 ) )
             => ~ ( ( ( ( produc8816143915943518219_c_nat @ P6 )
                      = X4 )
                   => ( Y4
                      = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ ( Xa @ ( produc2435110264752509257_c_nat @ P6 ) ) ) @ Ps2 ) ) )
                  & ( ( ( produc8816143915943518219_c_nat @ P6 )
                     != X4 )
                   => ( Y4
                      = ( cons_P7928565014262433479_c_nat @ P6 @ ( map_en7725529839661896901_c_nat @ X4 @ Xa @ Ps2 ) ) ) ) ) ) ) ) ).

% map_entry.elims
thf(fact_893_set__relcomp,axiom,
    ! [Xys: list_P7417839048565863355_c_nat,Yzs: list_P6797689553898924723_c_nat] :
      ( ( relcom2873162752072038412_c_nat @ ( set_Pr5807476769692288598_c_nat @ Xys ) @ ( set_Pr8775986980852786754_c_nat @ Yzs ) )
      = ( set_Pr5735414202003984296_c_nat
        @ ( concat2980991215343123196_c_nat
          @ ( map_Pr4432448568579428862_c_nat
            @ ^ [Xy: produc6689720525226317995_c_nat] :
                ( concat2980991215343123196_c_nat
                @ ( map_Pr2313767941087073910_c_nat
                  @ ^ [Yz: produc2591457285448022317_c_nat] :
                      ( if_lis6091463167075048391_c_nat
                      @ ( ( produc6319817161620345591_c_nat @ Xy )
                        = ( produc5231444489022403025_c_nat @ Yz ) )
                      @ ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( produc6033490199168946105_c_nat @ Xy ) @ ( produc7091416173205085715_c_nat @ Yz ) ) @ nil_Pr2407244426195543671_c_nat )
                      @ nil_Pr2407244426195543671_c_nat )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_894_set__relcomp,axiom,
    ! [Xys: list_P903359562653991662od_b_c,Yzs: list_P5760860764591065870_c_nat] :
      ( ( relcom6857592940063405457_c_nat @ ( set_Product_prod_b_c2 @ Xys ) @ ( set_Pr9108130572708629801_c_nat @ Yzs ) )
      = ( set_Pr5735414202003984296_c_nat
        @ ( concat2980991215343123196_c_nat
          @ ( map_Pr707172530603269169_c_nat
            @ ^ [Xy: product_prod_b_c] :
                ( concat2980991215343123196_c_nat
                @ ( map_Pr17017191815643857_c_nat
                  @ ^ [Yz: produc8234973081359154302_c_nat] :
                      ( if_lis6091463167075048391_c_nat
                      @ ( ( product_snd_b_c @ Xy )
                        = ( produc2965488249793387916_c_nat @ Yz ) )
                      @ ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( product_fst_b_c @ Xy ) @ ( produc5807826635457154762_c_nat @ Yz ) ) @ nil_Pr2407244426195543671_c_nat )
                      @ nil_Pr2407244426195543671_c_nat )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_895_set__relcomp,axiom,
    ! [Xys: list_P8155366108056616461_c_nat,Yzs: list_P2602636364039601413_c_nat] :
      ( ( relcom177692130347456094_c_nat @ ( set_Pr5735414202003984296_c_nat @ Xys ) @ ( set_Pr992668100555672468_c_nat @ Yzs ) )
      = ( set_Pr5735414202003984296_c_nat
        @ ( concat2980991215343123196_c_nat
          @ ( map_Pr8543597516576548176_c_nat
            @ ^ [Xy: produc4862256710654508797_c_nat] :
                ( concat2980991215343123196_c_nat
                @ ( map_Pr5374970998031937992_c_nat
                  @ ^ [Yz: produc3510132715374707071_c_nat] :
                      ( if_lis6091463167075048391_c_nat
                      @ ( ( produc2435110264752509257_c_nat @ Xy )
                        = ( produc8598682482351136035_c_nat @ Yz ) )
                      @ ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( produc8816143915943518219_c_nat @ Xy ) @ ( produc5611719534782831461_c_nat @ Yz ) ) @ nil_Pr2407244426195543671_c_nat )
                      @ nil_Pr2407244426195543671_c_nat )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_896_set__relcomp,axiom,
    ! [Xys: list_P832325522607645677od_b_b,Yzs: list_P8155366108056616461_c_nat] :
      ( ( relcom3484876569358759952_c_nat @ ( set_Product_prod_b_b2 @ Xys ) @ ( set_Pr5735414202003984296_c_nat @ Yzs ) )
      = ( set_Pr5735414202003984296_c_nat
        @ ( concat2980991215343123196_c_nat
          @ ( map_Pr6949367127585041584_c_nat
            @ ^ [Xy: product_prod_b_b] :
                ( concat2980991215343123196_c_nat
                @ ( map_Pr8543597516576548176_c_nat
                  @ ^ [Yz: produc4862256710654508797_c_nat] :
                      ( if_lis6091463167075048391_c_nat
                      @ ( ( product_snd_b_b @ Xy )
                        = ( produc8816143915943518219_c_nat @ Yz ) )
                      @ ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( product_fst_b_b @ Xy ) @ ( produc2435110264752509257_c_nat @ Yz ) ) @ nil_Pr2407244426195543671_c_nat )
                      @ nil_Pr2407244426195543671_c_nat )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_897_mmap__fold__lookup__rec,axiom,
    ! [R2: list_P5561950507774946575_c_nat,M: list_P5561950507774946575_c_nat,F: product_prod_c_nat > product_prod_c_nat,C: nat > nat > nat,Z: c] :
      ( ( distinct_c @ ( map_Pr72372295602292475_nat_c @ product_fst_c_nat @ R2 ) )
     => ( ( mmap_lookup_c_nat @ ( mmap_fold_c_nat @ M @ F @ C @ R2 ) @ Z )
        = ( case_l8977982442399753944at_nat @ ( mmap_lookup_c_nat @ R2 @ Z )
          @ ^ [V: nat,Vs: list_nat] :
              ( case_o7429725398727453821at_nat @ ( some_nat @ ( foldl_nat_nat @ C @ V @ Vs ) )
              @ ^ [W: nat] : ( some_nat @ ( foldl_nat_nat @ C @ W @ ( cons_nat @ V @ Vs ) ) )
              @ ( mmap_lookup_c_nat @ R2 @ Z ) )
          @ ( map_Pr5732458797000193813at_nat @ ( comp_P9136737662191791027_c_nat @ product_snd_c_nat @ F )
            @ ( filter2138311591596769474_c_nat
              @ ( produc7133942929724870258_nat_o
                @ ^ [K: c,V: nat] :
                    ( ( product_fst_c_nat @ ( F @ ( product_Pair_c_nat @ K @ V ) ) )
                    = Z ) )
              @ M ) ) ) ) ) ).

% mmap_fold_lookup_rec
thf(fact_898_mmap__fold__lookup__rec,axiom,
    ! [R2: list_P7417839048565863355_c_nat,M: list_P7417839048565863355_c_nat,F: produc6689720525226317995_c_nat > produc6689720525226317995_c_nat,C: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Z: b] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ R2 ) )
     => ( ( mmap_l5026999719965937591_c_nat @ ( mmap_f2405537622796709232_c_nat @ M @ F @ C @ R2 ) @ Z )
        = ( case_l293247108732037938_c_nat @ ( mmap_l5026999719965937591_c_nat @ R2 @ Z )
          @ ^ [V: produc4862256710654508797_c_nat,Vs: list_P8155366108056616461_c_nat] :
              ( case_o2363692632768546263_c_nat @ ( some_P2720002978653898840_c_nat @ ( foldl_7345954015341993117_c_nat @ C @ V @ Vs ) )
              @ ^ [W: produc4862256710654508797_c_nat] : ( some_P2720002978653898840_c_nat @ ( foldl_7345954015341993117_c_nat @ C @ W @ ( cons_P7928565014262433479_c_nat @ V @ Vs ) ) )
              @ ( mmap_l5026999719965937591_c_nat @ R2 @ Z ) )
          @ ( map_Pr4595408008933130094_c_nat @ ( comp_P5952376330372963768_c_nat @ produc6319817161620345591_c_nat @ F )
            @ ( filter3549032352921181294_c_nat
              @ ( produc6484547981188837958_nat_o
                @ ^ [K: b,V: produc4862256710654508797_c_nat] :
                    ( ( produc6033490199168946105_c_nat @ ( F @ ( produc8868828497325608613_c_nat @ K @ V ) ) )
                    = Z ) )
              @ M ) ) ) ) ) ).

% mmap_fold_lookup_rec
thf(fact_899_mmap__fold__lookup__rec,axiom,
    ! [R2: list_P903359562653991662od_b_c,M: list_P903359562653991662od_b_c,F: product_prod_b_c > product_prod_b_c,C: c > c > c,Z: b] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ R2 ) )
     => ( ( mmap_lookup_b_c @ ( mmap_fold_b_c @ M @ F @ C @ R2 ) @ Z )
        = ( case_list_option_c_c @ ( mmap_lookup_b_c @ R2 @ Z )
          @ ^ [V: c,Vs: list_c] :
              ( case_o1877681393294459589on_c_c @ ( some_c @ ( foldl_c_c @ C @ V @ Vs ) )
              @ ^ [W: c] : ( some_c @ ( foldl_c_c @ C @ W @ ( cons_c @ V @ Vs ) ) )
              @ ( mmap_lookup_b_c @ R2 @ Z ) )
          @ ( map_Pr8222292247188198876_b_c_c @ ( comp_P7269037405381514541od_b_c @ product_snd_b_c @ F )
            @ ( filter4794301956807392997od_b_c
              @ ( produc1873627588981547243_b_c_o
                @ ^ [K: b,V: c] :
                    ( ( product_fst_b_c @ ( F @ ( product_Pair_b_c @ K @ V ) ) )
                    = Z ) )
              @ M ) ) ) ) ) ).

% mmap_fold_lookup_rec
thf(fact_900_mmap__fold__lookup__rec,axiom,
    ! [R2: list_P8155366108056616461_c_nat,M: list_P8155366108056616461_c_nat,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,C: option7520157102916957007_c_nat > option7520157102916957007_c_nat > option7520157102916957007_c_nat,Z: b] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ R2 ) )
     => ( ( mmap_l1578203946870656521_c_nat @ ( mmap_f908472374020558018_c_nat @ M @ F @ C @ R2 ) @ Z )
        = ( case_l1252254761337325270_c_nat @ ( mmap_l1578203946870656521_c_nat @ R2 @ Z )
          @ ^ [V: option7520157102916957007_c_nat,Vs: list_o9079336047566018143_c_nat] :
              ( case_o7010352838590118779_c_nat @ ( some_o7552056839342673834_c_nat @ ( foldl_7092927554537613889_c_nat @ C @ V @ Vs ) )
              @ ^ [W: option7520157102916957007_c_nat] : ( some_o7552056839342673834_c_nat @ ( foldl_7092927554537613889_c_nat @ C @ W @ ( cons_o7152427081109263641_c_nat @ V @ Vs ) ) )
              @ ( mmap_l1578203946870656521_c_nat @ R2 @ Z ) )
          @ ( map_Pr8931211203828011794_c_nat @ ( comp_P4021412037527218862_c_nat @ produc2435110264752509257_c_nat @ F )
            @ ( filter4765930412304252352_c_nat
              @ ( produc2544960137906461044_nat_o
                @ ^ [K: b,V: option7520157102916957007_c_nat] :
                    ( ( produc8816143915943518219_c_nat @ ( F @ ( produc5716802255202478839_c_nat @ K @ V ) ) )
                    = Z ) )
              @ M ) ) ) ) ) ).

% mmap_fold_lookup_rec
thf(fact_901_set__filter,axiom,
    ! [P2: b > $o,Xs: list_b] :
      ( ( set_b2 @ ( filter_b @ P2 @ Xs ) )
      = ( collect_b
        @ ^ [X3: b] :
            ( ( member_b @ X3 @ ( set_b2 @ Xs ) )
            & ( P2 @ X3 ) ) ) ) ).

% set_filter
thf(fact_902_filter__map,axiom,
    ! [P2: b > $o,F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat] :
      ( ( filter_b @ P2 @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) )
      = ( map_Pr6200325787298368846_nat_b @ F @ ( filter3549032352921181294_c_nat @ ( comp_b9107354593259966439_c_nat @ P2 @ F ) @ Xs ) ) ) ).

% filter_map
thf(fact_903_filter__map,axiom,
    ! [P2: b > $o,F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( filter_b @ P2 @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) )
      = ( map_Pr8222292247188198875_b_c_b @ F @ ( filter4794301956807392997od_b_c @ ( comp_b2056900688142920748od_b_c @ P2 @ F ) @ Xs ) ) ) ).

% filter_map
thf(fact_904_relcomp_Ocases,axiom,
    ! [A1: b,A22: option7520157102916957007_c_nat,R2: set_Product_prod_b_b,S2: set_Pr8806432033423503795_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A1 @ A22 ) @ ( relcom3484876569358759952_c_nat @ R2 @ S2 ) )
     => ~ ! [B3: b] :
            ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A1 @ B3 ) @ R2 )
           => ~ ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ B3 @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_905_relcomp_Ocases,axiom,
    ! [A1: b,A22: option7520157102916957007_c_nat,R2: set_Pr8806432033423503795_c_nat,S2: set_Pr4914646408830571231_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A1 @ A22 ) @ ( relcom177692130347456094_c_nat @ R2 @ S2 ) )
     => ~ ! [B3: option7520157102916957007_c_nat] :
            ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A1 @ B3 ) @ R2 )
           => ~ ( member6296142003466748712_c_nat @ ( produc4206217144015867191_c_nat @ B3 @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_906_relcomp_Ocases,axiom,
    ! [A1: c,A22: nat,R2: set_Product_prod_c_c,S2: set_Pr6903500605879609269_c_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A1 @ A22 ) @ ( relcomp_c_c_nat @ R2 @ S2 ) )
     => ~ ! [B3: c] :
            ( ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ A1 @ B3 ) @ R2 )
           => ~ ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ B3 @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_907_relcomp_Ocases,axiom,
    ! [A1: c,A22: nat,R2: set_Pr6903500605879609269_c_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A1 @ A22 ) @ ( relcomp_c_nat_nat @ R2 @ S2 ) )
     => ~ ! [B3: nat] :
            ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A1 @ B3 ) @ R2 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_908_relcomp_Osimps,axiom,
    ! [A1: b,A22: option7520157102916957007_c_nat,R2: set_Product_prod_b_b,S2: set_Pr8806432033423503795_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A1 @ A22 ) @ ( relcom3484876569358759952_c_nat @ R2 @ S2 ) )
      = ( ? [A5: b,B4: b,C3: option7520157102916957007_c_nat] :
            ( ( A1 = A5 )
            & ( A22 = C3 )
            & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A5 @ B4 ) @ R2 )
            & ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ B4 @ C3 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_909_relcomp_Osimps,axiom,
    ! [A1: b,A22: option7520157102916957007_c_nat,R2: set_Pr8806432033423503795_c_nat,S2: set_Pr4914646408830571231_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A1 @ A22 ) @ ( relcom177692130347456094_c_nat @ R2 @ S2 ) )
      = ( ? [A5: b,B4: option7520157102916957007_c_nat,C3: option7520157102916957007_c_nat] :
            ( ( A1 = A5 )
            & ( A22 = C3 )
            & ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A5 @ B4 ) @ R2 )
            & ( member6296142003466748712_c_nat @ ( produc4206217144015867191_c_nat @ B4 @ C3 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_910_relcomp_Osimps,axiom,
    ! [A1: c,A22: nat,R2: set_Product_prod_c_c,S2: set_Pr6903500605879609269_c_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A1 @ A22 ) @ ( relcomp_c_c_nat @ R2 @ S2 ) )
      = ( ? [A5: c,B4: c,C3: nat] :
            ( ( A1 = A5 )
            & ( A22 = C3 )
            & ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ A5 @ B4 ) @ R2 )
            & ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ B4 @ C3 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_911_relcomp_Osimps,axiom,
    ! [A1: c,A22: nat,R2: set_Pr6903500605879609269_c_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A1 @ A22 ) @ ( relcomp_c_nat_nat @ R2 @ S2 ) )
      = ( ? [A5: c,B4: nat,C3: nat] :
            ( ( A1 = A5 )
            & ( A22 = C3 )
            & ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A5 @ B4 ) @ R2 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C3 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_912_relcomp_OrelcompI,axiom,
    ! [A: b,B: b,R2: set_Product_prod_b_b,C: option7520157102916957007_c_nat,S2: set_Pr8806432033423503795_c_nat] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
     => ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ B @ C ) @ S2 )
       => ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A @ C ) @ ( relcom3484876569358759952_c_nat @ R2 @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_913_relcomp_OrelcompI,axiom,
    ! [A: c,B: c,R2: set_Product_prod_c_c,C: nat,S2: set_Pr6903500605879609269_c_nat] :
      ( ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ A @ B ) @ R2 )
     => ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ B @ C ) @ S2 )
       => ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A @ C ) @ ( relcomp_c_c_nat @ R2 @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_914_relcomp_OrelcompI,axiom,
    ! [A: b,B: option7520157102916957007_c_nat,R2: set_Pr8806432033423503795_c_nat,C: option7520157102916957007_c_nat,S2: set_Pr4914646408830571231_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A @ B ) @ R2 )
     => ( ( member6296142003466748712_c_nat @ ( produc4206217144015867191_c_nat @ B @ C ) @ S2 )
       => ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A @ C ) @ ( relcom177692130347456094_c_nat @ R2 @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_915_relcomp_OrelcompI,axiom,
    ! [A: c,B: nat,R2: set_Pr6903500605879609269_c_nat,C: nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A @ B ) @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ S2 )
       => ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A @ C ) @ ( relcomp_c_nat_nat @ R2 @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_916_relcompE,axiom,
    ! [Xz: produc4862256710654508797_c_nat,R2: set_Product_prod_b_b,S2: set_Pr8806432033423503795_c_nat] :
      ( ( member7562873241046315796_c_nat @ Xz @ ( relcom3484876569358759952_c_nat @ R2 @ S2 ) )
     => ~ ! [X: b,Y: b,Z3: option7520157102916957007_c_nat] :
            ( ( Xz
              = ( produc5716802255202478839_c_nat @ X @ Z3 ) )
           => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Y ) @ R2 )
             => ~ ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ Y @ Z3 ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_917_relcompE,axiom,
    ! [Xz: produc4862256710654508797_c_nat,R2: set_Pr8806432033423503795_c_nat,S2: set_Pr4914646408830571231_c_nat] :
      ( ( member7562873241046315796_c_nat @ Xz @ ( relcom177692130347456094_c_nat @ R2 @ S2 ) )
     => ~ ! [X: b,Y: option7520157102916957007_c_nat,Z3: option7520157102916957007_c_nat] :
            ( ( Xz
              = ( produc5716802255202478839_c_nat @ X @ Z3 ) )
           => ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X @ Y ) @ R2 )
             => ~ ( member6296142003466748712_c_nat @ ( produc4206217144015867191_c_nat @ Y @ Z3 ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_918_relcompE,axiom,
    ! [Xz: product_prod_c_nat,R2: set_Product_prod_c_c,S2: set_Pr6903500605879609269_c_nat] :
      ( ( member8195077246299207702_c_nat @ Xz @ ( relcomp_c_c_nat @ R2 @ S2 ) )
     => ~ ! [X: c,Y: c,Z3: nat] :
            ( ( Xz
              = ( product_Pair_c_nat @ X @ Z3 ) )
           => ( ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ X @ Y ) @ R2 )
             => ~ ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ Y @ Z3 ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_919_relcompE,axiom,
    ! [Xz: product_prod_c_nat,R2: set_Pr6903500605879609269_c_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8195077246299207702_c_nat @ Xz @ ( relcomp_c_nat_nat @ R2 @ S2 ) )
     => ~ ! [X: c,Y: nat,Z3: nat] :
            ( ( Xz
              = ( product_Pair_c_nat @ X @ Z3 ) )
           => ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X @ Y ) @ R2 )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z3 ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_920_relcompEpair,axiom,
    ! [A: b,C: option7520157102916957007_c_nat,R2: set_Product_prod_b_b,S2: set_Pr8806432033423503795_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A @ C ) @ ( relcom3484876569358759952_c_nat @ R2 @ S2 ) )
     => ~ ! [B3: b] :
            ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B3 ) @ R2 )
           => ~ ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ B3 @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_921_relcompEpair,axiom,
    ! [A: b,C: option7520157102916957007_c_nat,R2: set_Pr8806432033423503795_c_nat,S2: set_Pr4914646408830571231_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A @ C ) @ ( relcom177692130347456094_c_nat @ R2 @ S2 ) )
     => ~ ! [B3: option7520157102916957007_c_nat] :
            ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ A @ B3 ) @ R2 )
           => ~ ( member6296142003466748712_c_nat @ ( produc4206217144015867191_c_nat @ B3 @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_922_relcompEpair,axiom,
    ! [A: c,C: nat,R2: set_Product_prod_c_c,S2: set_Pr6903500605879609269_c_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A @ C ) @ ( relcomp_c_c_nat @ R2 @ S2 ) )
     => ~ ! [B3: c] :
            ( ( member5074992359041316560od_c_c @ ( product_Pair_c_c @ A @ B3 ) @ R2 )
           => ~ ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ B3 @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_923_relcompEpair,axiom,
    ! [A: c,C: nat,R2: set_Pr6903500605879609269_c_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A @ C ) @ ( relcomp_c_nat_nat @ R2 @ S2 ) )
     => ~ ! [B3: nat] :
            ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ A @ B3 ) @ R2 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_924_distinct__map__filter,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,P2: produc6689720525226317995_c_nat > $o] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ F @ ( filter3549032352921181294_c_nat @ P2 @ Xs ) ) ) ) ).

% distinct_map_filter
thf(fact_925_distinct__map__filter,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,P2: product_prod_b_c > $o] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ F @ ( filter4794301956807392997od_b_c @ P2 @ Xs ) ) ) ) ).

% distinct_map_filter
thf(fact_926_map__ran__filter,axiom,
    ! [F: b > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,A: b,Ps: list_P7417839048565863355_c_nat] :
      ( ( map_ra2537877954580484484_c_nat @ F
        @ ( filter3549032352921181294_c_nat
          @ ^ [P5: produc6689720525226317995_c_nat] :
              ( ( produc6033490199168946105_c_nat @ P5 )
             != A )
          @ Ps ) )
      = ( filter3549032352921181294_c_nat
        @ ^ [P5: produc6689720525226317995_c_nat] :
            ( ( produc6033490199168946105_c_nat @ P5 )
           != A )
        @ ( map_ra2537877954580484484_c_nat @ F @ Ps ) ) ) ).

% map_ran_filter
thf(fact_927_map__ran__filter,axiom,
    ! [F: b > produc4862256710654508797_c_nat > c,A: b,Ps: list_P7417839048565863355_c_nat] :
      ( ( map_ra6186387241003446201_nat_c @ F
        @ ( filter3549032352921181294_c_nat
          @ ^ [P5: produc6689720525226317995_c_nat] :
              ( ( produc6033490199168946105_c_nat @ P5 )
             != A )
          @ Ps ) )
      = ( filter4794301956807392997od_b_c
        @ ^ [P5: product_prod_b_c] :
            ( ( product_fst_b_c @ P5 )
           != A )
        @ ( map_ra6186387241003446201_nat_c @ F @ Ps ) ) ) ).

% map_ran_filter
thf(fact_928_map__ran__filter,axiom,
    ! [F: b > produc4862256710654508797_c_nat > option7520157102916957007_c_nat,A: b,Ps: list_P7417839048565863355_c_nat] :
      ( ( map_ra3237904418225743062_c_nat @ F
        @ ( filter3549032352921181294_c_nat
          @ ^ [P5: produc6689720525226317995_c_nat] :
              ( ( produc6033490199168946105_c_nat @ P5 )
             != A )
          @ Ps ) )
      = ( filter4765930412304252352_c_nat
        @ ^ [P5: produc4862256710654508797_c_nat] :
            ( ( produc8816143915943518219_c_nat @ P5 )
           != A )
        @ ( map_ra3237904418225743062_c_nat @ F @ Ps ) ) ) ).

% map_ran_filter
thf(fact_929_map__ran__filter,axiom,
    ! [F: b > c > produc4862256710654508797_c_nat,A: b,Ps: list_P903359562653991662od_b_c] :
      ( ( map_ra6584430756116811765_c_nat @ F
        @ ( filter4794301956807392997od_b_c
          @ ^ [P5: product_prod_b_c] :
              ( ( product_fst_b_c @ P5 )
             != A )
          @ Ps ) )
      = ( filter3549032352921181294_c_nat
        @ ^ [P5: produc6689720525226317995_c_nat] :
            ( ( produc6033490199168946105_c_nat @ P5 )
           != A )
        @ ( map_ra6584430756116811765_c_nat @ F @ Ps ) ) ) ).

% map_ran_filter
thf(fact_930_map__ran__filter,axiom,
    ! [F: b > c > c,A: b,Ps: list_P903359562653991662od_b_c] :
      ( ( map_ran_b_c_c @ F
        @ ( filter4794301956807392997od_b_c
          @ ^ [P5: product_prod_b_c] :
              ( ( product_fst_b_c @ P5 )
             != A )
          @ Ps ) )
      = ( filter4794301956807392997od_b_c
        @ ^ [P5: product_prod_b_c] :
            ( ( product_fst_b_c @ P5 )
           != A )
        @ ( map_ran_b_c_c @ F @ Ps ) ) ) ).

% map_ran_filter
thf(fact_931_map__ran__filter,axiom,
    ! [F: b > c > option7520157102916957007_c_nat,A: b,Ps: list_P903359562653991662od_b_c] :
      ( ( map_ra2860756734228053063_c_nat @ F
        @ ( filter4794301956807392997od_b_c
          @ ^ [P5: product_prod_b_c] :
              ( ( product_fst_b_c @ P5 )
             != A )
          @ Ps ) )
      = ( filter4765930412304252352_c_nat
        @ ^ [P5: produc4862256710654508797_c_nat] :
            ( ( produc8816143915943518219_c_nat @ P5 )
           != A )
        @ ( map_ra2860756734228053063_c_nat @ F @ Ps ) ) ) ).

% map_ran_filter
thf(fact_932_map__ran__filter,axiom,
    ! [F: b > option7520157102916957007_c_nat > produc4862256710654508797_c_nat,A: b,Ps: list_P8155366108056616461_c_nat] :
      ( ( map_ra5752783577693071318_c_nat @ F
        @ ( filter4765930412304252352_c_nat
          @ ^ [P5: produc4862256710654508797_c_nat] :
              ( ( produc8816143915943518219_c_nat @ P5 )
             != A )
          @ Ps ) )
      = ( filter3549032352921181294_c_nat
        @ ^ [P5: produc6689720525226317995_c_nat] :
            ( ( produc6033490199168946105_c_nat @ P5 )
           != A )
        @ ( map_ra5752783577693071318_c_nat @ F @ Ps ) ) ) ).

% map_ran_filter
thf(fact_933_map__ran__filter,axiom,
    ! [F: b > option7520157102916957007_c_nat > c,A: b,Ps: list_P8155366108056616461_c_nat] :
      ( ( map_ra3617418671075910375_nat_c @ F
        @ ( filter4765930412304252352_c_nat
          @ ^ [P5: produc4862256710654508797_c_nat] :
              ( ( produc8816143915943518219_c_nat @ P5 )
             != A )
          @ Ps ) )
      = ( filter4794301956807392997od_b_c
        @ ^ [P5: product_prod_b_c] :
            ( ( product_fst_b_c @ P5 )
           != A )
        @ ( map_ra3617418671075910375_nat_c @ F @ Ps ) ) ) ).

% map_ran_filter
thf(fact_934_map__ran__filter,axiom,
    ! [F: b > option7520157102916957007_c_nat > option7520157102916957007_c_nat,A: b,Ps: list_P8155366108056616461_c_nat] :
      ( ( map_ra1379142644618337576_c_nat @ F
        @ ( filter4765930412304252352_c_nat
          @ ^ [P5: produc4862256710654508797_c_nat] :
              ( ( produc8816143915943518219_c_nat @ P5 )
             != A )
          @ Ps ) )
      = ( filter4765930412304252352_c_nat
        @ ^ [P5: produc4862256710654508797_c_nat] :
            ( ( produc8816143915943518219_c_nat @ P5 )
           != A )
        @ ( map_ra1379142644618337576_c_nat @ F @ Ps ) ) ) ).

% map_ran_filter
thf(fact_935_fst__diag__fst,axiom,
    ( ( comp_P8700490683832790374_c_nat @ product_fst_b_b
      @ ( comp_b2204499998293906230_c_nat
        @ ^ [X3: b] : ( product_Pair_b_b @ X3 @ X3 )
        @ produc6033490199168946105_c_nat ) )
    = produc6033490199168946105_c_nat ) ).

% fst_diag_fst
thf(fact_936_fst__diag__fst,axiom,
    ( ( comp_P2928534557564827629od_b_c @ product_fst_b_b
      @ ( comp_b3785825894137351709od_b_c
        @ ^ [X3: b] : ( product_Pair_b_b @ X3 @ X3 )
        @ product_fst_b_c ) )
    = product_fst_b_c ) ).

% fst_diag_fst
thf(fact_937_fst__diag__fst,axiom,
    ( ( comp_P5363440831588107960_c_nat @ product_fst_b_b
      @ ( comp_b6198840652669533320_c_nat
        @ ^ [X3: b] : ( product_Pair_b_b @ X3 @ X3 )
        @ produc8816143915943518219_c_nat ) )
    = produc8816143915943518219_c_nat ) ).

% fst_diag_fst
thf(fact_938_snd__diag__fst,axiom,
    ( ( comp_P8700490683832790374_c_nat @ product_snd_b_b
      @ ( comp_b2204499998293906230_c_nat
        @ ^ [X3: b] : ( product_Pair_b_b @ X3 @ X3 )
        @ produc6033490199168946105_c_nat ) )
    = produc6033490199168946105_c_nat ) ).

% snd_diag_fst
thf(fact_939_snd__diag__fst,axiom,
    ( ( comp_P2928534557564827629od_b_c @ product_snd_b_b
      @ ( comp_b3785825894137351709od_b_c
        @ ^ [X3: b] : ( product_Pair_b_b @ X3 @ X3 )
        @ product_fst_b_c ) )
    = product_fst_b_c ) ).

% snd_diag_fst
thf(fact_940_snd__diag__fst,axiom,
    ( ( comp_P5363440831588107960_c_nat @ product_snd_b_b
      @ ( comp_b6198840652669533320_c_nat
        @ ^ [X3: b] : ( product_Pair_b_b @ X3 @ X3 )
        @ produc8816143915943518219_c_nat ) )
    = produc8816143915943518219_c_nat ) ).

% snd_diag_fst
thf(fact_941_distinct__map__entry,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,K2: b,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( map_en8198299921509431923_c_nat @ K2 @ F @ Xs ) ) ) ) ).

% distinct_map_entry
thf(fact_942_distinct__map__entry,axiom,
    ! [Xs: list_P903359562653991662od_b_c,K2: b,F: c > c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( map_entry_b_c @ K2 @ F @ Xs ) ) ) ) ).

% distinct_map_entry
thf(fact_943_distinct__map__entry,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,K2: b,F: option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs ) )
     => ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( map_en7725529839661896901_c_nat @ K2 @ F @ Xs ) ) ) ) ).

% distinct_map_entry
thf(fact_944_map__of__map,axiom,
    ! [F: option7520157102916957007_c_nat > option7520157102916957007_c_nat,Xs: list_P8155366108056616461_c_nat] :
      ( ( map_of2598500733885366699_c_nat
        @ ( map_Pr5967270692098401216_c_nat
          @ ( produc3722540595886809633_c_nat
            @ ^ [K: b,V: option7520157102916957007_c_nat] : ( produc5716802255202478839_c_nat @ K @ ( F @ V ) ) )
          @ Xs ) )
      = ( comp_o2788451701313185436_nat_b @ ( map_op5348336799008523307_c_nat @ F ) @ ( map_of2598500733885366699_c_nat @ Xs ) ) ) ).

% map_of_map
thf(fact_945_dom__map__entry,axiom,
    ! [K2: b,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Xs: list_P7417839048565863355_c_nat] :
      ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ ( map_en8198299921509431923_c_nat @ K2 @ F @ Xs ) ) )
      = ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Xs ) ) ) ).

% dom_map_entry
thf(fact_946_dom__map__entry,axiom,
    ! [K2: b,F: c > c,Xs: list_P903359562653991662od_b_c] :
      ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( map_entry_b_c @ K2 @ F @ Xs ) ) )
      = ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Xs ) ) ) ).

% dom_map_entry
thf(fact_947_dom__map__entry,axiom,
    ! [K2: b,F: option7520157102916957007_c_nat > option7520157102916957007_c_nat,Xs: list_P8155366108056616461_c_nat] :
      ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ ( map_en7725529839661896901_c_nat @ K2 @ F @ Xs ) ) )
      = ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Xs ) ) ) ).

% dom_map_entry
thf(fact_948_map__entry_Osimps_I2_J,axiom,
    ! [P: product_prod_c_nat,K2: c,F: nat > nat,Ps: list_P5561950507774946575_c_nat] :
      ( ( ( ( product_fst_c_nat @ P )
          = K2 )
       => ( ( map_entry_c_nat @ K2 @ F @ ( cons_P7676055461599297353_c_nat @ P @ Ps ) )
          = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K2 @ ( F @ ( product_snd_c_nat @ P ) ) ) @ Ps ) ) )
      & ( ( ( product_fst_c_nat @ P )
         != K2 )
       => ( ( map_entry_c_nat @ K2 @ F @ ( cons_P7676055461599297353_c_nat @ P @ Ps ) )
          = ( cons_P7676055461599297353_c_nat @ P @ ( map_entry_c_nat @ K2 @ F @ Ps ) ) ) ) ) ).

% map_entry.simps(2)
thf(fact_949_map__entry_Osimps_I2_J,axiom,
    ! [P: produc6689720525226317995_c_nat,K2: b,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Ps: list_P7417839048565863355_c_nat] :
      ( ( ( ( produc6033490199168946105_c_nat @ P )
          = K2 )
       => ( ( map_en8198299921509431923_c_nat @ K2 @ F @ ( cons_P7294201633519847029_c_nat @ P @ Ps ) )
          = ( cons_P7294201633519847029_c_nat @ ( produc8868828497325608613_c_nat @ K2 @ ( F @ ( produc6319817161620345591_c_nat @ P ) ) ) @ Ps ) ) )
      & ( ( ( produc6033490199168946105_c_nat @ P )
         != K2 )
       => ( ( map_en8198299921509431923_c_nat @ K2 @ F @ ( cons_P7294201633519847029_c_nat @ P @ Ps ) )
          = ( cons_P7294201633519847029_c_nat @ P @ ( map_en8198299921509431923_c_nat @ K2 @ F @ Ps ) ) ) ) ) ).

% map_entry.simps(2)
thf(fact_950_map__entry_Osimps_I2_J,axiom,
    ! [P: product_prod_b_c,K2: b,F: c > c,Ps: list_P903359562653991662od_b_c] :
      ( ( ( ( product_fst_b_c @ P )
          = K2 )
       => ( ( map_entry_b_c @ K2 @ F @ ( cons_P4529483553340347422od_b_c @ P @ Ps ) )
          = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ K2 @ ( F @ ( product_snd_b_c @ P ) ) ) @ Ps ) ) )
      & ( ( ( product_fst_b_c @ P )
         != K2 )
       => ( ( map_entry_b_c @ K2 @ F @ ( cons_P4529483553340347422od_b_c @ P @ Ps ) )
          = ( cons_P4529483553340347422od_b_c @ P @ ( map_entry_b_c @ K2 @ F @ Ps ) ) ) ) ) ).

% map_entry.simps(2)
thf(fact_951_map__entry_Osimps_I2_J,axiom,
    ! [P: produc4862256710654508797_c_nat,K2: b,F: option7520157102916957007_c_nat > option7520157102916957007_c_nat,Ps: list_P8155366108056616461_c_nat] :
      ( ( ( ( produc8816143915943518219_c_nat @ P )
          = K2 )
       => ( ( map_en7725529839661896901_c_nat @ K2 @ F @ ( cons_P7928565014262433479_c_nat @ P @ Ps ) )
          = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ ( F @ ( produc2435110264752509257_c_nat @ P ) ) ) @ Ps ) ) )
      & ( ( ( produc8816143915943518219_c_nat @ P )
         != K2 )
       => ( ( map_en7725529839661896901_c_nat @ K2 @ F @ ( cons_P7928565014262433479_c_nat @ P @ Ps ) )
          = ( cons_P7928565014262433479_c_nat @ P @ ( map_en7725529839661896901_c_nat @ K2 @ F @ Ps ) ) ) ) ) ).

% map_entry.simps(2)
thf(fact_952_mmap__fold__lookup,axiom,
    ! [M: list_P5561950507774946575_c_nat,F: product_prod_c_nat > product_prod_c_nat,C: nat > nat > nat,Z: c] :
      ( ( distinct_c @ ( map_Pr72372295602292475_nat_c @ product_fst_c_nat @ M ) )
     => ( ( mmap_lookup_c_nat @ ( mmap_fold_c_nat @ M @ F @ C @ nil_Pr650041864559007481_c_nat ) @ Z )
        = ( case_l8977982442399753944at_nat @ none_nat
          @ ^ [V: nat,Vs: list_nat] : ( some_nat @ ( foldl_nat_nat @ C @ V @ Vs ) )
          @ ( map_Pr5732458797000193813at_nat @ ( comp_P9136737662191791027_c_nat @ product_snd_c_nat @ F )
            @ ( filter2138311591596769474_c_nat
              @ ( produc7133942929724870258_nat_o
                @ ^ [K: c,V: nat] :
                    ( ( product_fst_c_nat @ ( F @ ( product_Pair_c_nat @ K @ V ) ) )
                    = Z ) )
              @ M ) ) ) ) ) ).

% mmap_fold_lookup
thf(fact_953_mmap__fold__lookup,axiom,
    ! [M: list_P7417839048565863355_c_nat,F: produc6689720525226317995_c_nat > produc6689720525226317995_c_nat,C: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Z: b] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ M ) )
     => ( ( mmap_l5026999719965937591_c_nat @ ( mmap_f2405537622796709232_c_nat @ M @ F @ C @ nil_Pr7000167559216756261_c_nat ) @ Z )
        = ( case_l293247108732037938_c_nat @ none_P8487441334512977628_c_nat
          @ ^ [V: produc4862256710654508797_c_nat,Vs: list_P8155366108056616461_c_nat] : ( some_P2720002978653898840_c_nat @ ( foldl_7345954015341993117_c_nat @ C @ V @ Vs ) )
          @ ( map_Pr4595408008933130094_c_nat @ ( comp_P5952376330372963768_c_nat @ produc6319817161620345591_c_nat @ F )
            @ ( filter3549032352921181294_c_nat
              @ ( produc6484547981188837958_nat_o
                @ ^ [K: b,V: produc4862256710654508797_c_nat] :
                    ( ( produc6033490199168946105_c_nat @ ( F @ ( produc8868828497325608613_c_nat @ K @ V ) ) )
                    = Z ) )
              @ M ) ) ) ) ) ).

% mmap_fold_lookup
thf(fact_954_mmap__fold__lookup,axiom,
    ! [M: list_P903359562653991662od_b_c,F: product_prod_b_c > product_prod_b_c,C: c > c > c,Z: b] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ M ) )
     => ( ( mmap_lookup_b_c @ ( mmap_fold_b_c @ M @ F @ C @ nil_Product_prod_b_c ) @ Z )
        = ( case_list_option_c_c @ none_c
          @ ^ [V: c,Vs: list_c] : ( some_c @ ( foldl_c_c @ C @ V @ Vs ) )
          @ ( map_Pr8222292247188198876_b_c_c @ ( comp_P7269037405381514541od_b_c @ product_snd_b_c @ F )
            @ ( filter4794301956807392997od_b_c
              @ ( produc1873627588981547243_b_c_o
                @ ^ [K: b,V: c] :
                    ( ( product_fst_b_c @ ( F @ ( product_Pair_b_c @ K @ V ) ) )
                    = Z ) )
              @ M ) ) ) ) ) ).

% mmap_fold_lookup
thf(fact_955_mmap__fold__lookup,axiom,
    ! [M: list_P8155366108056616461_c_nat,F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,C: option7520157102916957007_c_nat > option7520157102916957007_c_nat > option7520157102916957007_c_nat,Z: b] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ M ) )
     => ( ( mmap_l1578203946870656521_c_nat @ ( mmap_f908472374020558018_c_nat @ M @ F @ C @ nil_Pr2407244426195543671_c_nat ) @ Z )
        = ( case_l1252254761337325270_c_nat @ none_o1329286346417566254_c_nat
          @ ^ [V: option7520157102916957007_c_nat,Vs: list_o9079336047566018143_c_nat] : ( some_o7552056839342673834_c_nat @ ( foldl_7092927554537613889_c_nat @ C @ V @ Vs ) )
          @ ( map_Pr8931211203828011794_c_nat @ ( comp_P4021412037527218862_c_nat @ produc2435110264752509257_c_nat @ F )
            @ ( filter4765930412304252352_c_nat
              @ ( produc2544960137906461044_nat_o
                @ ^ [K: b,V: option7520157102916957007_c_nat] :
                    ( ( produc8816143915943518219_c_nat @ ( F @ ( produc5716802255202478839_c_nat @ K @ V ) ) )
                    = Z ) )
              @ M ) ) ) ) ) ).

% mmap_fold_lookup
thf(fact_956_lookup__fold__sup,axiom,
    ! [M: list_P903359562653991662od_b_c,F: b > b,Z: b] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ M ) )
     => ( ( mmap_lookup_b_c @ ( fold_sup_b_c @ M @ F ) @ Z )
        = ( if_option_c
          @ ( ( collect_b
              @ ^ [X3: b] :
                  ( ( member_b @ X3 @ ( mmap_keys_b_c @ M ) )
                  & ( ( F @ X3 )
                    = Z ) ) )
            = bot_bot_set_b )
          @ none_c
          @ ( some_c
            @ ( lattic6792493950031347383_fin_c
              @ ( image_b_c @ ( comp_option_c_c_b @ the_c @ ( mmap_lookup_b_c @ M ) )
                @ ( collect_b
                  @ ^ [X3: b] :
                      ( ( member_b @ X3 @ ( mmap_keys_b_c @ M ) )
                      & ( ( F @ X3 )
                        = Z ) ) ) ) ) ) ) ) ) ).

% lookup_fold_sup
thf(fact_957_map__filter__def,axiom,
    ( map_fi174849983179987656_nat_b
    = ( ^ [F3: produc6689720525226317995_c_nat > option_b,Xs2: list_P7417839048565863355_c_nat] :
          ( map_Pr6200325787298368846_nat_b @ ( comp_o2219867182051211400_c_nat @ the_b @ F3 )
          @ ( filter3549032352921181294_c_nat
            @ ^ [X3: produc6689720525226317995_c_nat] :
                ( ( F3 @ X3 )
               != none_b )
            @ Xs2 ) ) ) ) ).

% map_filter_def
thf(fact_958_map__filter__def,axiom,
    ( map_fi694150099910991841_b_c_b
    = ( ^ [F3: product_prod_b_c > option_b,Xs2: list_P903359562653991662od_b_c] :
          ( map_Pr8222292247188198875_b_c_b @ ( comp_o7848714244194021963od_b_c @ the_b @ F3 )
          @ ( filter4794301956807392997od_b_c
            @ ^ [X3: product_prod_b_c] :
                ( ( F3 @ X3 )
               != none_b )
            @ Xs2 ) ) ) ) ).

% map_filter_def
thf(fact_959_map__filter__map__filter,axiom,
    ! [F: produc6689720525226317995_c_nat > b,P2: produc6689720525226317995_c_nat > $o,Xs: list_P7417839048565863355_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ F @ ( filter3549032352921181294_c_nat @ P2 @ Xs ) )
      = ( map_fi174849983179987656_nat_b
        @ ^ [X3: produc6689720525226317995_c_nat] : ( if_option_b @ ( P2 @ X3 ) @ ( some_b @ ( F @ X3 ) ) @ none_b )
        @ Xs ) ) ).

% map_filter_map_filter
thf(fact_960_map__filter__map__filter,axiom,
    ! [F: product_prod_b_c > b,P2: product_prod_b_c > $o,Xs: list_P903359562653991662od_b_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( filter4794301956807392997od_b_c @ P2 @ Xs ) )
      = ( map_fi694150099910991841_b_c_b
        @ ^ [X3: product_prod_b_c] : ( if_option_b @ ( P2 @ X3 ) @ ( some_b @ ( F @ X3 ) ) @ none_b )
        @ Xs ) ) ).

% map_filter_map_filter
thf(fact_961_map__default_Opelims,axiom,
    ! [X4: c,Xa: nat,Xb: nat > nat,Xc: list_P5561950507774946575_c_nat,Y4: list_P5561950507774946575_c_nat] :
      ( ( ( map_default_c_nat @ X4 @ Xa @ Xb @ Xc )
        = Y4 )
     => ( ( accp_P1911109574288323642_c_nat @ map_de3077052858721673583_c_nat @ ( produc7314949712312174461_c_nat @ X4 @ ( produc5798563371203896454_c_nat @ Xa @ ( produc1941097459371650279_c_nat @ Xb @ Xc ) ) ) )
       => ( ( ( Xc = nil_Pr650041864559007481_c_nat )
           => ( ( Y4
                = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ X4 @ Xa ) @ nil_Pr650041864559007481_c_nat ) )
             => ~ ( accp_P1911109574288323642_c_nat @ map_de3077052858721673583_c_nat @ ( produc7314949712312174461_c_nat @ X4 @ ( produc5798563371203896454_c_nat @ Xa @ ( produc1941097459371650279_c_nat @ Xb @ nil_Pr650041864559007481_c_nat ) ) ) ) ) )
         => ~ ! [P6: product_prod_c_nat,Ps2: list_P5561950507774946575_c_nat] :
                ( ( Xc
                  = ( cons_P7676055461599297353_c_nat @ P6 @ Ps2 ) )
               => ( ( ( ( ( product_fst_c_nat @ P6 )
                        = X4 )
                     => ( Y4
                        = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ X4 @ ( Xb @ ( product_snd_c_nat @ P6 ) ) ) @ Ps2 ) ) )
                    & ( ( ( product_fst_c_nat @ P6 )
                       != X4 )
                     => ( Y4
                        = ( cons_P7676055461599297353_c_nat @ P6 @ ( map_default_c_nat @ X4 @ Xa @ Xb @ Ps2 ) ) ) ) )
                 => ~ ( accp_P1911109574288323642_c_nat @ map_de3077052858721673583_c_nat @ ( produc7314949712312174461_c_nat @ X4 @ ( produc5798563371203896454_c_nat @ Xa @ ( produc1941097459371650279_c_nat @ Xb @ ( cons_P7676055461599297353_c_nat @ P6 @ Ps2 ) ) ) ) ) ) ) ) ) ) ).

% map_default.pelims
thf(fact_962_map__default_Opelims,axiom,
    ! [X4: b,Xa: produc4862256710654508797_c_nat,Xb: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Xc: list_P7417839048565863355_c_nat,Y4: list_P7417839048565863355_c_nat] :
      ( ( ( map_de8565690227408816292_c_nat @ X4 @ Xa @ Xb @ Xc )
        = Y4 )
     => ( ( accp_P6393395294101418450_c_nat @ map_de2601993647352999835_c_nat @ ( produc6689170455415217557_c_nat @ X4 @ ( produc7240219008752950559_c_nat @ Xa @ ( produc4003268613017673069_c_nat @ Xb @ Xc ) ) ) )
       => ( ( ( Xc = nil_Pr7000167559216756261_c_nat )
           => ( ( Y4
                = ( cons_P7294201633519847029_c_nat @ ( produc8868828497325608613_c_nat @ X4 @ Xa ) @ nil_Pr7000167559216756261_c_nat ) )
             => ~ ( accp_P6393395294101418450_c_nat @ map_de2601993647352999835_c_nat @ ( produc6689170455415217557_c_nat @ X4 @ ( produc7240219008752950559_c_nat @ Xa @ ( produc4003268613017673069_c_nat @ Xb @ nil_Pr7000167559216756261_c_nat ) ) ) ) ) )
         => ~ ! [P6: produc6689720525226317995_c_nat,Ps2: list_P7417839048565863355_c_nat] :
                ( ( Xc
                  = ( cons_P7294201633519847029_c_nat @ P6 @ Ps2 ) )
               => ( ( ( ( ( produc6033490199168946105_c_nat @ P6 )
                        = X4 )
                     => ( Y4
                        = ( cons_P7294201633519847029_c_nat @ ( produc8868828497325608613_c_nat @ X4 @ ( Xb @ ( produc6319817161620345591_c_nat @ P6 ) ) ) @ Ps2 ) ) )
                    & ( ( ( produc6033490199168946105_c_nat @ P6 )
                       != X4 )
                     => ( Y4
                        = ( cons_P7294201633519847029_c_nat @ P6 @ ( map_de8565690227408816292_c_nat @ X4 @ Xa @ Xb @ Ps2 ) ) ) ) )
                 => ~ ( accp_P6393395294101418450_c_nat @ map_de2601993647352999835_c_nat @ ( produc6689170455415217557_c_nat @ X4 @ ( produc7240219008752950559_c_nat @ Xa @ ( produc4003268613017673069_c_nat @ Xb @ ( cons_P7294201633519847029_c_nat @ P6 @ Ps2 ) ) ) ) ) ) ) ) ) ) ).

% map_default.pelims
thf(fact_963_map__default_Opelims,axiom,
    ! [X4: b,Xa: c,Xb: c > c,Xc: list_P903359562653991662od_b_c,Y4: list_P903359562653991662od_b_c] :
      ( ( ( map_default_b_c @ X4 @ Xa @ Xb @ Xc )
        = Y4 )
     => ( ( accp_P2383248136236608210od_b_c @ map_default_rel_b_c @ ( produc1384939585547926677od_b_c @ X4 @ ( produc3590379597927054951od_b_c @ Xa @ ( produc1101678828523717360od_b_c @ Xb @ Xc ) ) ) )
       => ( ( ( Xc = nil_Product_prod_b_c )
           => ( ( Y4
                = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ Xa ) @ nil_Product_prod_b_c ) )
             => ~ ( accp_P2383248136236608210od_b_c @ map_default_rel_b_c @ ( produc1384939585547926677od_b_c @ X4 @ ( produc3590379597927054951od_b_c @ Xa @ ( produc1101678828523717360od_b_c @ Xb @ nil_Product_prod_b_c ) ) ) ) ) )
         => ~ ! [P6: product_prod_b_c,Ps2: list_P903359562653991662od_b_c] :
                ( ( Xc
                  = ( cons_P4529483553340347422od_b_c @ P6 @ Ps2 ) )
               => ( ( ( ( ( product_fst_b_c @ P6 )
                        = X4 )
                     => ( Y4
                        = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ ( Xb @ ( product_snd_b_c @ P6 ) ) ) @ Ps2 ) ) )
                    & ( ( ( product_fst_b_c @ P6 )
                       != X4 )
                     => ( Y4
                        = ( cons_P4529483553340347422od_b_c @ P6 @ ( map_default_b_c @ X4 @ Xa @ Xb @ Ps2 ) ) ) ) )
                 => ~ ( accp_P2383248136236608210od_b_c @ map_default_rel_b_c @ ( produc1384939585547926677od_b_c @ X4 @ ( produc3590379597927054951od_b_c @ Xa @ ( produc1101678828523717360od_b_c @ Xb @ ( cons_P4529483553340347422od_b_c @ P6 @ Ps2 ) ) ) ) ) ) ) ) ) ) ).

% map_default.pelims
thf(fact_964_map__default_Opelims,axiom,
    ! [X4: b,Xa: option7520157102916957007_c_nat,Xb: option7520157102916957007_c_nat > option7520157102916957007_c_nat,Xc: list_P8155366108056616461_c_nat,Y4: list_P8155366108056616461_c_nat] :
      ( ( ( map_de5305620854455122934_c_nat @ X4 @ Xa @ Xb @ Xc )
        = Y4 )
     => ( ( accp_P3076908095482526198_c_nat @ map_de7178281694451429869_c_nat @ ( produc3000800644307087289_c_nat @ X4 @ ( produc8618114333783736899_c_nat @ Xa @ ( produc5195976547672773731_c_nat @ Xb @ Xc ) ) ) )
       => ( ( ( Xc = nil_Pr2407244426195543671_c_nat )
           => ( ( Y4
                = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Xa ) @ nil_Pr2407244426195543671_c_nat ) )
             => ~ ( accp_P3076908095482526198_c_nat @ map_de7178281694451429869_c_nat @ ( produc3000800644307087289_c_nat @ X4 @ ( produc8618114333783736899_c_nat @ Xa @ ( produc5195976547672773731_c_nat @ Xb @ nil_Pr2407244426195543671_c_nat ) ) ) ) ) )
         => ~ ! [P6: produc4862256710654508797_c_nat,Ps2: list_P8155366108056616461_c_nat] :
                ( ( Xc
                  = ( cons_P7928565014262433479_c_nat @ P6 @ Ps2 ) )
               => ( ( ( ( ( produc8816143915943518219_c_nat @ P6 )
                        = X4 )
                     => ( Y4
                        = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ ( Xb @ ( produc2435110264752509257_c_nat @ P6 ) ) ) @ Ps2 ) ) )
                    & ( ( ( produc8816143915943518219_c_nat @ P6 )
                       != X4 )
                     => ( Y4
                        = ( cons_P7928565014262433479_c_nat @ P6 @ ( map_de5305620854455122934_c_nat @ X4 @ Xa @ Xb @ Ps2 ) ) ) ) )
                 => ~ ( accp_P3076908095482526198_c_nat @ map_de7178281694451429869_c_nat @ ( produc3000800644307087289_c_nat @ X4 @ ( produc8618114333783736899_c_nat @ Xa @ ( produc5195976547672773731_c_nat @ Xb @ ( cons_P7928565014262433479_c_nat @ P6 @ Ps2 ) ) ) ) ) ) ) ) ) ) ).

% map_default.pelims
thf(fact_965_zip__Cons1,axiom,
    ! [X4: b,Xs: list_b,Ys: list_o9079336047566018143_c_nat] :
      ( ( zip_b_1039741285442616279_c_nat @ ( cons_b @ X4 @ Xs ) @ Ys )
      = ( case_l355151514845609284_c_nat @ nil_Pr2407244426195543671_c_nat
        @ ^ [Y3: option7520157102916957007_c_nat,Ys2: list_o9079336047566018143_c_nat] : ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y3 ) @ ( zip_b_1039741285442616279_c_nat @ Xs @ Ys2 ) )
        @ Ys ) ) ).

% zip_Cons1
thf(fact_966_zip__Cons1,axiom,
    ! [X4: c,Xs: list_c,Ys: list_nat] :
      ( ( zip_c_nat @ ( cons_c @ X4 @ Xs ) @ Ys )
      = ( case_l8321934174081100487at_nat @ nil_Pr650041864559007481_c_nat
        @ ^ [Y3: nat,Ys2: list_nat] : ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ X4 @ Y3 ) @ ( zip_c_nat @ Xs @ Ys2 ) )
        @ Ys ) ) ).

% zip_Cons1
thf(fact_967_zip__Cons,axiom,
    ! [Xs: list_b,Y4: option7520157102916957007_c_nat,Ys: list_o9079336047566018143_c_nat] :
      ( ( zip_b_1039741285442616279_c_nat @ Xs @ ( cons_o7152427081109263641_c_nat @ Y4 @ Ys ) )
      = ( case_l8175337382496304074_nat_b @ nil_Pr2407244426195543671_c_nat
        @ ^ [Z4: b,Zs3: list_b] : ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ Z4 @ Y4 ) @ ( zip_b_1039741285442616279_c_nat @ Zs3 @ Ys ) )
        @ Xs ) ) ).

% zip_Cons
thf(fact_968_zip__Cons,axiom,
    ! [Xs: list_c,Y4: nat,Ys: list_nat] :
      ( ( zip_c_nat @ Xs @ ( cons_nat @ Y4 @ Ys ) )
      = ( case_l3020224243550723017_nat_c @ nil_Pr650041864559007481_c_nat
        @ ^ [Z4: c,Zs3: list_c] : ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ Z4 @ Y4 ) @ ( zip_c_nat @ Zs3 @ Ys ) )
        @ Xs ) ) ).

% zip_Cons
thf(fact_969_map__entry_Opelims,axiom,
    ! [X4: c,Xa: nat > nat,Xb: list_P5561950507774946575_c_nat,Y4: list_P5561950507774946575_c_nat] :
      ( ( ( map_entry_c_nat @ X4 @ Xa @ Xb )
        = Y4 )
     => ( ( accp_P905144024986759903_c_nat @ map_entry_rel_c_nat @ ( produc17256813457310534_c_nat @ X4 @ ( produc1941097459371650279_c_nat @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_Pr650041864559007481_c_nat )
           => ( ( Y4 = nil_Pr650041864559007481_c_nat )
             => ~ ( accp_P905144024986759903_c_nat @ map_entry_rel_c_nat @ ( produc17256813457310534_c_nat @ X4 @ ( produc1941097459371650279_c_nat @ Xa @ nil_Pr650041864559007481_c_nat ) ) ) ) )
         => ~ ! [P6: product_prod_c_nat,Ps2: list_P5561950507774946575_c_nat] :
                ( ( Xb
                  = ( cons_P7676055461599297353_c_nat @ P6 @ Ps2 ) )
               => ( ( ( ( ( product_fst_c_nat @ P6 )
                        = X4 )
                     => ( Y4
                        = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ X4 @ ( Xa @ ( product_snd_c_nat @ P6 ) ) ) @ Ps2 ) ) )
                    & ( ( ( product_fst_c_nat @ P6 )
                       != X4 )
                     => ( Y4
                        = ( cons_P7676055461599297353_c_nat @ P6 @ ( map_entry_c_nat @ X4 @ Xa @ Ps2 ) ) ) ) )
                 => ~ ( accp_P905144024986759903_c_nat @ map_entry_rel_c_nat @ ( produc17256813457310534_c_nat @ X4 @ ( produc1941097459371650279_c_nat @ Xa @ ( cons_P7676055461599297353_c_nat @ P6 @ Ps2 ) ) ) ) ) ) ) ) ) ).

% map_entry.pelims
thf(fact_970_map__entry_Opelims,axiom,
    ! [X4: b,Xa: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Xb: list_P7417839048565863355_c_nat,Y4: list_P7417839048565863355_c_nat] :
      ( ( ( map_en8198299921509431923_c_nat @ X4 @ Xa @ Xb )
        = Y4 )
     => ( ( accp_P4202521573542701350_c_nat @ map_en4627508037358870668_c_nat @ ( produc1358758444348770189_c_nat @ X4 @ ( produc4003268613017673069_c_nat @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_Pr7000167559216756261_c_nat )
           => ( ( Y4 = nil_Pr7000167559216756261_c_nat )
             => ~ ( accp_P4202521573542701350_c_nat @ map_en4627508037358870668_c_nat @ ( produc1358758444348770189_c_nat @ X4 @ ( produc4003268613017673069_c_nat @ Xa @ nil_Pr7000167559216756261_c_nat ) ) ) ) )
         => ~ ! [P6: produc6689720525226317995_c_nat,Ps2: list_P7417839048565863355_c_nat] :
                ( ( Xb
                  = ( cons_P7294201633519847029_c_nat @ P6 @ Ps2 ) )
               => ( ( ( ( ( produc6033490199168946105_c_nat @ P6 )
                        = X4 )
                     => ( Y4
                        = ( cons_P7294201633519847029_c_nat @ ( produc8868828497325608613_c_nat @ X4 @ ( Xa @ ( produc6319817161620345591_c_nat @ P6 ) ) ) @ Ps2 ) ) )
                    & ( ( ( produc6033490199168946105_c_nat @ P6 )
                       != X4 )
                     => ( Y4
                        = ( cons_P7294201633519847029_c_nat @ P6 @ ( map_en8198299921509431923_c_nat @ X4 @ Xa @ Ps2 ) ) ) ) )
                 => ~ ( accp_P4202521573542701350_c_nat @ map_en4627508037358870668_c_nat @ ( produc1358758444348770189_c_nat @ X4 @ ( produc4003268613017673069_c_nat @ Xa @ ( cons_P7294201633519847029_c_nat @ P6 @ Ps2 ) ) ) ) ) ) ) ) ) ).

% map_entry.pelims
thf(fact_971_map__entry_Opelims,axiom,
    ! [X4: b,Xa: c > c,Xb: list_P903359562653991662od_b_c,Y4: list_P903359562653991662od_b_c] :
      ( ( ( map_entry_b_c @ X4 @ Xa @ Xb )
        = Y4 )
     => ( ( accp_P4462124054252585635od_b_c @ map_entry_rel_b_c @ ( produc7015427828931889894od_b_c @ X4 @ ( produc1101678828523717360od_b_c @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_Product_prod_b_c )
           => ( ( Y4 = nil_Product_prod_b_c )
             => ~ ( accp_P4462124054252585635od_b_c @ map_entry_rel_b_c @ ( produc7015427828931889894od_b_c @ X4 @ ( produc1101678828523717360od_b_c @ Xa @ nil_Product_prod_b_c ) ) ) ) )
         => ~ ! [P6: product_prod_b_c,Ps2: list_P903359562653991662od_b_c] :
                ( ( Xb
                  = ( cons_P4529483553340347422od_b_c @ P6 @ Ps2 ) )
               => ( ( ( ( ( product_fst_b_c @ P6 )
                        = X4 )
                     => ( Y4
                        = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ X4 @ ( Xa @ ( product_snd_b_c @ P6 ) ) ) @ Ps2 ) ) )
                    & ( ( ( product_fst_b_c @ P6 )
                       != X4 )
                     => ( Y4
                        = ( cons_P4529483553340347422od_b_c @ P6 @ ( map_entry_b_c @ X4 @ Xa @ Ps2 ) ) ) ) )
                 => ~ ( accp_P4462124054252585635od_b_c @ map_entry_rel_b_c @ ( produc7015427828931889894od_b_c @ X4 @ ( produc1101678828523717360od_b_c @ Xa @ ( cons_P4529483553340347422od_b_c @ P6 @ Ps2 ) ) ) ) ) ) ) ) ) ).

% map_entry.pelims
thf(fact_972_map__entry_Opelims,axiom,
    ! [X4: b,Xa: option7520157102916957007_c_nat > option7520157102916957007_c_nat,Xb: list_P8155366108056616461_c_nat,Y4: list_P8155366108056616461_c_nat] :
      ( ( ( map_en7725529839661896901_c_nat @ X4 @ Xa @ Xb )
        = Y4 )
     => ( ( accp_P993573161634354460_c_nat @ map_en4477831636919090142_c_nat @ ( produc1008487875849483395_c_nat @ X4 @ ( produc5195976547672773731_c_nat @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_Pr2407244426195543671_c_nat )
           => ( ( Y4 = nil_Pr2407244426195543671_c_nat )
             => ~ ( accp_P993573161634354460_c_nat @ map_en4477831636919090142_c_nat @ ( produc1008487875849483395_c_nat @ X4 @ ( produc5195976547672773731_c_nat @ Xa @ nil_Pr2407244426195543671_c_nat ) ) ) ) )
         => ~ ! [P6: produc4862256710654508797_c_nat,Ps2: list_P8155366108056616461_c_nat] :
                ( ( Xb
                  = ( cons_P7928565014262433479_c_nat @ P6 @ Ps2 ) )
               => ( ( ( ( ( produc8816143915943518219_c_nat @ P6 )
                        = X4 )
                     => ( Y4
                        = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ ( Xa @ ( produc2435110264752509257_c_nat @ P6 ) ) ) @ Ps2 ) ) )
                    & ( ( ( produc8816143915943518219_c_nat @ P6 )
                       != X4 )
                     => ( Y4
                        = ( cons_P7928565014262433479_c_nat @ P6 @ ( map_en7725529839661896901_c_nat @ X4 @ Xa @ Ps2 ) ) ) ) )
                 => ~ ( accp_P993573161634354460_c_nat @ map_en4477831636919090142_c_nat @ ( produc1008487875849483395_c_nat @ X4 @ ( produc5195976547672773731_c_nat @ Xa @ ( cons_P7928565014262433479_c_nat @ P6 @ Ps2 ) ) ) ) ) ) ) ) ) ).

% map_entry.pelims
thf(fact_973_delete__aux_Opelims,axiom,
    ! [X4: b,Xa: list_P8155366108056616461_c_nat,Y4: list_P8155366108056616461_c_nat] :
      ( ( ( delete7176321171147170984_c_nat @ X4 @ Xa )
        = Y4 )
     => ( ( accp_P2639456933007735538_c_nat @ delete3416536959023130555_c_nat @ ( produc385356081875622325_c_nat @ X4 @ Xa ) )
       => ( ( ( Xa = nil_Pr2407244426195543671_c_nat )
           => ( ( Y4 = nil_Pr2407244426195543671_c_nat )
             => ~ ( accp_P2639456933007735538_c_nat @ delete3416536959023130555_c_nat @ ( produc385356081875622325_c_nat @ X4 @ nil_Pr2407244426195543671_c_nat ) ) ) )
         => ~ ! [K4: b,V3: option7520157102916957007_c_nat,Xs4: list_P8155366108056616461_c_nat] :
                ( ( Xa
                  = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K4 @ V3 ) @ Xs4 ) )
               => ( ( ( ( X4 = K4 )
                     => ( Y4 = Xs4 ) )
                    & ( ( X4 != K4 )
                     => ( Y4
                        = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K4 @ V3 ) @ ( delete7176321171147170984_c_nat @ X4 @ Xs4 ) ) ) ) )
                 => ~ ( accp_P2639456933007735538_c_nat @ delete3416536959023130555_c_nat @ ( produc385356081875622325_c_nat @ X4 @ ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K4 @ V3 ) @ Xs4 ) ) ) ) ) ) ) ) ).

% delete_aux.pelims
thf(fact_974_delete__aux_Opelims,axiom,
    ! [X4: c,Xa: list_P5561950507774946575_c_nat,Y4: list_P5561950507774946575_c_nat] :
      ( ( ( delete_aux_c_nat @ X4 @ Xa )
        = Y4 )
     => ( ( accp_P2110349762303447157_c_nat @ delete_aux_rel_c_nat @ ( produc2615319020942666552_c_nat @ X4 @ Xa ) )
       => ( ( ( Xa = nil_Pr650041864559007481_c_nat )
           => ( ( Y4 = nil_Pr650041864559007481_c_nat )
             => ~ ( accp_P2110349762303447157_c_nat @ delete_aux_rel_c_nat @ ( produc2615319020942666552_c_nat @ X4 @ nil_Pr650041864559007481_c_nat ) ) ) )
         => ~ ! [K4: c,V3: nat,Xs4: list_P5561950507774946575_c_nat] :
                ( ( Xa
                  = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K4 @ V3 ) @ Xs4 ) )
               => ( ( ( ( X4 = K4 )
                     => ( Y4 = Xs4 ) )
                    & ( ( X4 != K4 )
                     => ( Y4
                        = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K4 @ V3 ) @ ( delete_aux_c_nat @ X4 @ Xs4 ) ) ) ) )
                 => ~ ( accp_P2110349762303447157_c_nat @ delete_aux_rel_c_nat @ ( produc2615319020942666552_c_nat @ X4 @ ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K4 @ V3 ) @ Xs4 ) ) ) ) ) ) ) ) ).

% delete_aux.pelims
thf(fact_975_map__of__map__restrict,axiom,
    ! [F: b > option7520157102916957007_c_nat,Ks: list_b] :
      ( ( map_of2598500733885366699_c_nat
        @ ( map_b_6442276761558007352_c_nat
          @ ^ [K: b] : ( produc5716802255202478839_c_nat @ K @ ( F @ K ) )
          @ Ks ) )
      = ( restri8862894144314794270_c_nat @ ( comp_o1392410879592115052_nat_b @ some_o7552056839342673834_c_nat @ F ) @ ( set_b2 @ Ks ) ) ) ).

% map_of_map_restrict
thf(fact_976_map__of__map__restrict,axiom,
    ! [F: c > nat,Ks: list_c] :
      ( ( map_of_c_nat
        @ ( map_c_3232936969464635451_c_nat
          @ ^ [K: c] : ( product_Pair_c_nat @ K @ ( F @ K ) )
          @ Ks ) )
      = ( restrict_map_c_nat @ ( comp_n4566197705232572779_nat_c @ some_nat @ F ) @ ( set_c2 @ Ks ) ) ) ).

% map_of_map_restrict
thf(fact_977_restrict__out,axiom,
    ! [X4: b,A4: set_b,M: b > option2970150418924381261_c_nat] :
      ( ~ ( member_b @ X4 @ A4 )
     => ( ( restri4430789530154927564_c_nat @ M @ A4 @ X4 )
        = none_P8487441334512977628_c_nat ) ) ).

% restrict_out
thf(fact_978_restrict__map__def,axiom,
    ( restri4430789530154927564_c_nat
    = ( ^ [M2: b > option2970150418924381261_c_nat,A6: set_b,X3: b] : ( if_opt6485458775533684743_c_nat @ ( member_b @ X3 @ A6 ) @ ( M2 @ X3 ) @ none_P8487441334512977628_c_nat ) ) ) ).

% restrict_map_def
thf(fact_979_transpose__map__map,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_l7485416851377393739_c_nat] :
      ( ( transpose_b @ ( map_li1180621075770707268list_b @ ( map_Pr6200325787298368846_nat_b @ F ) @ Xs ) )
      = ( map_li1180621075770707268list_b @ ( map_Pr6200325787298368846_nat_b @ F ) @ ( transp983579826141871387_c_nat @ Xs ) ) ) ).

% transpose_map_map
thf(fact_980_transpose__map__map,axiom,
    ! [F: product_prod_b_c > b,Xs: list_l8907847357763382004od_b_c] :
      ( ( transpose_b @ ( map_li1600214420876558683list_b @ ( map_Pr8222292247188198875_b_c_b @ F ) @ Xs ) )
      = ( map_li1600214420876558683list_b @ ( map_Pr8222292247188198875_b_c_b @ F ) @ ( transp1488944918686653944od_b_c @ Xs ) ) ) ).

% transpose_map_map
thf(fact_981_delete__id,axiom,
    ! [K2: b,Al: list_P7417839048565863355_c_nat] :
      ( ~ ( member_b @ K2 @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Al ) ) )
     => ( ( delete1755679665865935934_c_nat @ K2 @ Al )
        = Al ) ) ).

% delete_id
thf(fact_982_delete__id,axiom,
    ! [K2: b,Al: list_P903359562653991662od_b_c] :
      ( ~ ( member_b @ K2 @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Al ) ) )
     => ( ( delete_b_c @ K2 @ Al )
        = Al ) ) ).

% delete_id
thf(fact_983_delete__id,axiom,
    ! [K2: b,Al: list_P8155366108056616461_c_nat] :
      ( ~ ( member_b @ K2 @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Al ) ) )
     => ( ( delete4993458036327213456_c_nat @ K2 @ Al )
        = Al ) ) ).

% delete_id
thf(fact_984_map__tl,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ F @ ( tl_Pro3829623603210221948_c_nat @ Xs ) )
      = ( tl_b @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) ) ) ).

% map_tl
thf(fact_985_map__tl,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( tl_Product_prod_b_c @ Xs ) )
      = ( tl_b @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) ) ) ).

% map_tl
thf(fact_986_hd__zip,axiom,
    ! [Xs: list_b,Ys: list_o9079336047566018143_c_nat] :
      ( ( Xs != nil_b )
     => ( ( Ys != nil_op2410232952894710985_c_nat )
       => ( ( hd_Pro908356766978074578_c_nat @ ( zip_b_1039741285442616279_c_nat @ Xs @ Ys ) )
          = ( produc5716802255202478839_c_nat @ ( hd_b @ Xs ) @ ( hd_opt1486276835331225892_c_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_987_hd__zip,axiom,
    ! [Xs: list_c,Ys: list_nat] :
      ( ( Xs != nil_c )
     => ( ( Ys != nil_nat )
       => ( ( hd_Pro2182721878771191764_c_nat @ ( zip_c_nat @ Xs @ Ys ) )
          = ( product_Pair_c_nat @ ( hd_c @ Xs ) @ ( hd_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_988_list_Omap__sel_I1_J,axiom,
    ! [A: list_P7417839048565863355_c_nat,F: produc6689720525226317995_c_nat > b] :
      ( ( A != nil_Pr7000167559216756261_c_nat )
     => ( ( hd_b @ ( map_Pr6200325787298368846_nat_b @ F @ A ) )
        = ( F @ ( hd_Pro890923934664041600_c_nat @ A ) ) ) ) ).

% list.map_sel(1)
thf(fact_989_list_Omap__sel_I1_J,axiom,
    ! [A: list_P903359562653991662od_b_c,F: product_prod_b_c > b] :
      ( ( A != nil_Product_prod_b_c )
     => ( ( hd_b @ ( map_Pr8222292247188198875_b_c_b @ F @ A ) )
        = ( F @ ( hd_Product_prod_b_c @ A ) ) ) ) ).

% list.map_sel(1)
thf(fact_990_hd__map,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,F: produc6689720525226317995_c_nat > b] :
      ( ( Xs != nil_Pr7000167559216756261_c_nat )
     => ( ( hd_b @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) )
        = ( F @ ( hd_Pro890923934664041600_c_nat @ Xs ) ) ) ) ).

% hd_map
thf(fact_991_hd__map,axiom,
    ! [Xs: list_P903359562653991662od_b_c,F: product_prod_b_c > b] :
      ( ( Xs != nil_Product_prod_b_c )
     => ( ( hd_b @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) )
        = ( F @ ( hd_Product_prod_b_c @ Xs ) ) ) ) ).

% hd_map
thf(fact_992_list_Omap__sel_I2_J,axiom,
    ! [A: list_P7417839048565863355_c_nat,F: produc6689720525226317995_c_nat > b] :
      ( ( A != nil_Pr7000167559216756261_c_nat )
     => ( ( tl_b @ ( map_Pr6200325787298368846_nat_b @ F @ A ) )
        = ( map_Pr6200325787298368846_nat_b @ F @ ( tl_Pro3829623603210221948_c_nat @ A ) ) ) ) ).

% list.map_sel(2)
thf(fact_993_list_Omap__sel_I2_J,axiom,
    ! [A: list_P903359562653991662od_b_c,F: product_prod_b_c > b] :
      ( ( A != nil_Product_prod_b_c )
     => ( ( tl_b @ ( map_Pr8222292247188198875_b_c_b @ F @ A ) )
        = ( map_Pr8222292247188198875_b_c_b @ F @ ( tl_Product_prod_b_c @ A ) ) ) ) ).

% list.map_sel(2)
thf(fact_994_delete__map,axiom,
    ! [F: produc6689720525226317995_c_nat > produc6689720525226317995_c_nat,K2: b,Ps: list_P7417839048565863355_c_nat] :
      ( ! [Kv: produc6689720525226317995_c_nat] :
          ( ( produc6033490199168946105_c_nat @ ( F @ Kv ) )
          = ( produc6033490199168946105_c_nat @ Kv ) )
     => ( ( delete1755679665865935934_c_nat @ K2 @ ( map_Pr7976378254695727132_c_nat @ F @ Ps ) )
        = ( map_Pr7976378254695727132_c_nat @ F @ ( delete1755679665865935934_c_nat @ K2 @ Ps ) ) ) ) ).

% delete_map
thf(fact_995_delete__map,axiom,
    ! [F: product_prod_b_c > produc6689720525226317995_c_nat,K2: b,Ps: list_P903359562653991662od_b_c] :
      ( ! [Kv: product_prod_b_c] :
          ( ( produc6033490199168946105_c_nat @ ( F @ Kv ) )
          = ( product_fst_b_c @ Kv ) )
     => ( ( delete1755679665865935934_c_nat @ K2 @ ( map_Pr6155953585549333711_c_nat @ F @ Ps ) )
        = ( map_Pr6155953585549333711_c_nat @ F @ ( delete_b_c @ K2 @ Ps ) ) ) ) ).

% delete_map
thf(fact_996_delete__map,axiom,
    ! [F: produc4862256710654508797_c_nat > produc6689720525226317995_c_nat,K2: b,Ps: list_P8155366108056616461_c_nat] :
      ( ! [Kv: produc4862256710654508797_c_nat] :
          ( ( produc6033490199168946105_c_nat @ ( F @ Kv ) )
          = ( produc8816143915943518219_c_nat @ Kv ) )
     => ( ( delete1755679665865935934_c_nat @ K2 @ ( map_Pr5604758227450724974_c_nat @ F @ Ps ) )
        = ( map_Pr5604758227450724974_c_nat @ F @ ( delete4993458036327213456_c_nat @ K2 @ Ps ) ) ) ) ).

% delete_map
thf(fact_997_delete__map,axiom,
    ! [F: produc6689720525226317995_c_nat > product_prod_b_c,K2: b,Ps: list_P7417839048565863355_c_nat] :
      ( ! [Kv: produc6689720525226317995_c_nat] :
          ( ( product_fst_b_c @ ( F @ Kv ) )
          = ( produc6033490199168946105_c_nat @ Kv ) )
     => ( ( delete_b_c @ K2 @ ( map_Pr5949920527225541047od_b_c @ F @ Ps ) )
        = ( map_Pr5949920527225541047od_b_c @ F @ ( delete1755679665865935934_c_nat @ K2 @ Ps ) ) ) ) ).

% delete_map
thf(fact_998_delete__map,axiom,
    ! [F: product_prod_b_c > product_prod_b_c,K2: b,Ps: list_P903359562653991662od_b_c] :
      ( ! [Kv: product_prod_b_c] :
          ( ( product_fst_b_c @ ( F @ Kv ) )
          = ( product_fst_b_c @ Kv ) )
     => ( ( delete_b_c @ K2 @ ( map_Pr6384634858812015940od_b_c @ F @ Ps ) )
        = ( map_Pr6384634858812015940od_b_c @ F @ ( delete_b_c @ K2 @ Ps ) ) ) ) ).

% delete_map
thf(fact_999_delete__map,axiom,
    ! [F: produc4862256710654508797_c_nat > product_prod_b_c,K2: b,Ps: list_P8155366108056616461_c_nat] :
      ( ! [Kv: produc4862256710654508797_c_nat] :
          ( ( product_fst_b_c @ ( F @ Kv ) )
          = ( produc8816143915943518219_c_nat @ Kv ) )
     => ( ( delete_b_c @ K2 @ ( map_Pr8130720921054730469od_b_c @ F @ Ps ) )
        = ( map_Pr8130720921054730469od_b_c @ F @ ( delete4993458036327213456_c_nat @ K2 @ Ps ) ) ) ) ).

% delete_map
thf(fact_1000_delete__map,axiom,
    ! [F: produc6689720525226317995_c_nat > produc4862256710654508797_c_nat,K2: b,Ps: list_P7417839048565863355_c_nat] :
      ( ! [Kv: produc6689720525226317995_c_nat] :
          ( ( produc8816143915943518219_c_nat @ ( F @ Kv ) )
          = ( produc6033490199168946105_c_nat @ Kv ) )
     => ( ( delete4993458036327213456_c_nat @ K2 @ ( map_Pr4595408008933130094_c_nat @ F @ Ps ) )
        = ( map_Pr4595408008933130094_c_nat @ F @ ( delete1755679665865935934_c_nat @ K2 @ Ps ) ) ) ) ).

% delete_map
thf(fact_1001_delete__map,axiom,
    ! [F: product_prod_b_c > produc4862256710654508797_c_nat,K2: b,Ps: list_P903359562653991662od_b_c] :
      ( ! [Kv: product_prod_b_c] :
          ( ( produc8816143915943518219_c_nat @ ( F @ Kv ) )
          = ( product_fst_b_c @ Kv ) )
     => ( ( delete4993458036327213456_c_nat @ K2 @ ( map_Pr2101552211129450273_c_nat @ F @ Ps ) )
        = ( map_Pr2101552211129450273_c_nat @ F @ ( delete_b_c @ K2 @ Ps ) ) ) ) ).

% delete_map
thf(fact_1002_delete__map,axiom,
    ! [F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,K2: b,Ps: list_P8155366108056616461_c_nat] :
      ( ! [Kv: produc4862256710654508797_c_nat] :
          ( ( produc8816143915943518219_c_nat @ ( F @ Kv ) )
          = ( produc8816143915943518219_c_nat @ Kv ) )
     => ( ( delete4993458036327213456_c_nat @ K2 @ ( map_Pr5967270692098401216_c_nat @ F @ Ps ) )
        = ( map_Pr5967270692098401216_c_nat @ F @ ( delete4993458036327213456_c_nat @ K2 @ Ps ) ) ) ) ).

% delete_map
thf(fact_1003_distinct__delete,axiom,
    ! [Al: list_P7417839048565863355_c_nat,K2: b] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( delete1755679665865935934_c_nat @ K2 @ Al ) ) ) ) ).

% distinct_delete
thf(fact_1004_distinct__delete,axiom,
    ! [Al: list_P903359562653991662od_b_c,K2: b] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( delete_b_c @ K2 @ Al ) ) ) ) ).

% distinct_delete
thf(fact_1005_distinct__delete,axiom,
    ! [Al: list_P8155366108056616461_c_nat,K2: b] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) )
     => ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( delete4993458036327213456_c_nat @ K2 @ Al ) ) ) ) ).

% distinct_delete
thf(fact_1006_delete__notin__dom,axiom,
    ! [K2: b,Al: list_P7417839048565863355_c_nat] :
      ~ ( member_b @ K2 @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ ( delete1755679665865935934_c_nat @ K2 @ Al ) ) ) ) ).

% delete_notin_dom
thf(fact_1007_delete__notin__dom,axiom,
    ! [K2: b,Al: list_P903359562653991662od_b_c] :
      ~ ( member_b @ K2 @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( delete_b_c @ K2 @ Al ) ) ) ) ).

% delete_notin_dom
thf(fact_1008_delete__notin__dom,axiom,
    ! [K2: b,Al: list_P8155366108056616461_c_nat] :
      ~ ( member_b @ K2 @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ ( delete4993458036327213456_c_nat @ K2 @ Al ) ) ) ) ).

% delete_notin_dom
thf(fact_1009_compose_Opinduct,axiom,
    ! [A0: list_P7417839048565863355_c_nat,A1: list_P5405424656961669729_c_nat,P2: list_P7417839048565863355_c_nat > list_P5405424656961669729_c_nat > $o] :
      ( ( accp_P628919224718545170_c_nat @ compos4366374410145781152_c_nat @ ( produc5158760630865638669_c_nat @ A0 @ A1 ) )
     => ( ! [Ys4: list_P5405424656961669729_c_nat] :
            ( ( accp_P628919224718545170_c_nat @ compos4366374410145781152_c_nat @ ( produc5158760630865638669_c_nat @ nil_Pr7000167559216756261_c_nat @ Ys4 ) )
           => ( P2 @ nil_Pr7000167559216756261_c_nat @ Ys4 ) )
       => ( ! [X: produc6689720525226317995_c_nat,Xs4: list_P7417839048565863355_c_nat,Ys4: list_P5405424656961669729_c_nat] :
              ( ( accp_P628919224718545170_c_nat @ compos4366374410145781152_c_nat @ ( produc5158760630865638669_c_nat @ ( cons_P7294201633519847029_c_nat @ X @ Xs4 ) @ Ys4 ) )
             => ( ( ( ( map_of2917391071775207903_c_nat @ Ys4 @ ( produc6319817161620345591_c_nat @ X ) )
                    = none_P8487441334512977628_c_nat )
                 => ( P2 @ ( delete1755679665865935934_c_nat @ ( produc6033490199168946105_c_nat @ X ) @ Xs4 ) @ Ys4 ) )
               => ( ! [X24: produc4862256710654508797_c_nat] :
                      ( ( ( map_of2917391071775207903_c_nat @ Ys4 @ ( produc6319817161620345591_c_nat @ X ) )
                        = ( some_P2720002978653898840_c_nat @ X24 ) )
                     => ( P2 @ Xs4 @ Ys4 ) )
                 => ( P2 @ ( cons_P7294201633519847029_c_nat @ X @ Xs4 ) @ Ys4 ) ) ) )
         => ( P2 @ A0 @ A1 ) ) ) ) ).

% compose.pinduct
thf(fact_1010_compose_Opinduct,axiom,
    ! [A0: list_P903359562653991662od_b_c,A1: list_P4207096184057010108_c_nat,P2: list_P903359562653991662od_b_c > list_P4207096184057010108_c_nat > $o] :
      ( ( accp_P2003788528859879184_c_nat @ compos1189029223378847961_c_nat @ ( produc4465663410034776147_c_nat @ A0 @ A1 ) )
     => ( ! [Ys4: list_P4207096184057010108_c_nat] :
            ( ( accp_P2003788528859879184_c_nat @ compos1189029223378847961_c_nat @ ( produc4465663410034776147_c_nat @ nil_Product_prod_b_c @ Ys4 ) )
           => ( P2 @ nil_Product_prod_b_c @ Ys4 ) )
       => ( ! [X: product_prod_b_c,Xs4: list_P903359562653991662od_b_c,Ys4: list_P4207096184057010108_c_nat] :
              ( ( accp_P2003788528859879184_c_nat @ compos1189029223378847961_c_nat @ ( produc4465663410034776147_c_nat @ ( cons_P4529483553340347422od_b_c @ X @ Xs4 ) @ Ys4 ) )
             => ( ( ( ( map_of4154038134980830682_c_nat @ Ys4 @ ( product_snd_b_c @ X ) )
                    = none_P8487441334512977628_c_nat )
                 => ( P2 @ ( delete_b_c @ ( product_fst_b_c @ X ) @ Xs4 ) @ Ys4 ) )
               => ( ! [X24: produc4862256710654508797_c_nat] :
                      ( ( ( map_of4154038134980830682_c_nat @ Ys4 @ ( product_snd_b_c @ X ) )
                        = ( some_P2720002978653898840_c_nat @ X24 ) )
                     => ( P2 @ Xs4 @ Ys4 ) )
                 => ( P2 @ ( cons_P4529483553340347422od_b_c @ X @ Xs4 ) @ Ys4 ) ) ) )
         => ( P2 @ A0 @ A1 ) ) ) ) ).

% compose.pinduct
thf(fact_1011_compose_Opinduct,axiom,
    ! [A0: list_P8155366108056616461_c_nat,A1: list_P2361265211252882099_c_nat,P2: list_P8155366108056616461_c_nat > list_P2361265211252882099_c_nat > $o] :
      ( ( accp_P8536316450249730194_c_nat @ compos3214990791252568050_c_nat @ ( produc142894216116831629_c_nat @ A0 @ A1 ) )
     => ( ! [Ys4: list_P2361265211252882099_c_nat] :
            ( ( accp_P8536316450249730194_c_nat @ compos3214990791252568050_c_nat @ ( produc142894216116831629_c_nat @ nil_Pr2407244426195543671_c_nat @ Ys4 ) )
           => ( P2 @ nil_Pr2407244426195543671_c_nat @ Ys4 ) )
       => ( ! [X: produc4862256710654508797_c_nat,Xs4: list_P8155366108056616461_c_nat,Ys4: list_P2361265211252882099_c_nat] :
              ( ( accp_P8536316450249730194_c_nat @ compos3214990791252568050_c_nat @ ( produc142894216116831629_c_nat @ ( cons_P7928565014262433479_c_nat @ X @ Xs4 ) @ Ys4 ) )
             => ( ( ( ( map_of2974102585088269105_c_nat @ Ys4 @ ( produc2435110264752509257_c_nat @ X ) )
                    = none_P8487441334512977628_c_nat )
                 => ( P2 @ ( delete4993458036327213456_c_nat @ ( produc8816143915943518219_c_nat @ X ) @ Xs4 ) @ Ys4 ) )
               => ( ! [X24: produc4862256710654508797_c_nat] :
                      ( ( ( map_of2974102585088269105_c_nat @ Ys4 @ ( produc2435110264752509257_c_nat @ X ) )
                        = ( some_P2720002978653898840_c_nat @ X24 ) )
                     => ( P2 @ Xs4 @ Ys4 ) )
                 => ( P2 @ ( cons_P7928565014262433479_c_nat @ X @ Xs4 ) @ Ys4 ) ) ) )
         => ( P2 @ A0 @ A1 ) ) ) ) ).

% compose.pinduct
thf(fact_1012_compose_Oelims,axiom,
    ! [X4: list_P7417839048565863355_c_nat,Xa: list_P6797689553898924723_c_nat,Y4: list_P8155366108056616461_c_nat] :
      ( ( ( compos3540800493588629485_c_nat @ X4 @ Xa )
        = Y4 )
     => ( ( ( X4 = nil_Pr7000167559216756261_c_nat )
         => ( Y4 != nil_Pr2407244426195543671_c_nat ) )
       => ~ ! [X: produc6689720525226317995_c_nat,Xs4: list_P7417839048565863355_c_nat] :
              ( ( X4
                = ( cons_P7294201633519847029_c_nat @ X @ Xs4 ) )
             => ( Y4
               != ( case_o2058764249211701737_c_nat @ ( compos3540800493588629485_c_nat @ ( delete1755679665865935934_c_nat @ ( produc6033490199168946105_c_nat @ X ) @ Xs4 ) @ Xa )
                  @ ^ [V: option7520157102916957007_c_nat] : ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( produc6033490199168946105_c_nat @ X ) @ V ) @ ( compos3540800493588629485_c_nat @ Xs4 @ Xa ) )
                  @ ( map_of459223425620940849_c_nat @ Xa @ ( produc6319817161620345591_c_nat @ X ) ) ) ) ) ) ) ).

% compose.elims
thf(fact_1013_compose_Oelims,axiom,
    ! [X4: list_P903359562653991662od_b_c,Xa: list_P5760860764591065870_c_nat,Y4: list_P8155366108056616461_c_nat] :
      ( ( ( compos1359844409926042736_c_nat @ X4 @ Xa )
        = Y4 )
     => ( ( ( X4 = nil_Product_prod_b_c )
         => ( Y4 != nil_Pr2407244426195543671_c_nat ) )
       => ~ ! [X: product_prod_b_c,Xs4: list_P903359562653991662od_b_c] :
              ( ( X4
                = ( cons_P4529483553340347422od_b_c @ X @ Xs4 ) )
             => ( Y4
               != ( case_o2058764249211701737_c_nat @ ( compos1359844409926042736_c_nat @ ( delete_b_c @ ( product_fst_b_c @ X ) @ Xs4 ) @ Xa )
                  @ ^ [V: option7520157102916957007_c_nat] : ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( product_fst_b_c @ X ) @ V ) @ ( compos1359844409926042736_c_nat @ Xs4 @ Xa ) )
                  @ ( map_of5971217104590012204_c_nat @ Xa @ ( product_snd_b_c @ X ) ) ) ) ) ) ) ).

% compose.elims
thf(fact_1014_compose_Oelims,axiom,
    ! [X4: list_P8155366108056616461_c_nat,Xa: list_P2602636364039601413_c_nat,Y4: list_P8155366108056616461_c_nat] :
      ( ( ( compos1451257573582057279_c_nat @ X4 @ Xa )
        = Y4 )
     => ( ( ( X4 = nil_Pr2407244426195543671_c_nat )
         => ( Y4 != nil_Pr2407244426195543671_c_nat ) )
       => ~ ! [X: produc4862256710654508797_c_nat,Xs4: list_P8155366108056616461_c_nat] :
              ( ( X4
                = ( cons_P7928565014262433479_c_nat @ X @ Xs4 ) )
             => ( Y4
               != ( case_o2058764249211701737_c_nat @ ( compos1451257573582057279_c_nat @ ( delete4993458036327213456_c_nat @ ( produc8816143915943518219_c_nat @ X ) @ Xs4 ) @ Xa )
                  @ ^ [V: option7520157102916957007_c_nat] : ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( produc8816143915943518219_c_nat @ X ) @ V ) @ ( compos1451257573582057279_c_nat @ Xs4 @ Xa ) )
                  @ ( map_of1110926660169950083_c_nat @ Xa @ ( produc2435110264752509257_c_nat @ X ) ) ) ) ) ) ) ).

% compose.elims
thf(fact_1015_compose_Osimps_I2_J,axiom,
    ! [X4: produc6689720525226317995_c_nat,Xs: list_P7417839048565863355_c_nat,Ys: list_P6797689553898924723_c_nat] :
      ( ( compos3540800493588629485_c_nat @ ( cons_P7294201633519847029_c_nat @ X4 @ Xs ) @ Ys )
      = ( case_o2058764249211701737_c_nat @ ( compos3540800493588629485_c_nat @ ( delete1755679665865935934_c_nat @ ( produc6033490199168946105_c_nat @ X4 ) @ Xs ) @ Ys )
        @ ^ [V: option7520157102916957007_c_nat] : ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( produc6033490199168946105_c_nat @ X4 ) @ V ) @ ( compos3540800493588629485_c_nat @ Xs @ Ys ) )
        @ ( map_of459223425620940849_c_nat @ Ys @ ( produc6319817161620345591_c_nat @ X4 ) ) ) ) ).

% compose.simps(2)
thf(fact_1016_compose_Osimps_I2_J,axiom,
    ! [X4: product_prod_b_c,Xs: list_P903359562653991662od_b_c,Ys: list_P5760860764591065870_c_nat] :
      ( ( compos1359844409926042736_c_nat @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs ) @ Ys )
      = ( case_o2058764249211701737_c_nat @ ( compos1359844409926042736_c_nat @ ( delete_b_c @ ( product_fst_b_c @ X4 ) @ Xs ) @ Ys )
        @ ^ [V: option7520157102916957007_c_nat] : ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( product_fst_b_c @ X4 ) @ V ) @ ( compos1359844409926042736_c_nat @ Xs @ Ys ) )
        @ ( map_of5971217104590012204_c_nat @ Ys @ ( product_snd_b_c @ X4 ) ) ) ) ).

% compose.simps(2)
thf(fact_1017_compose_Osimps_I2_J,axiom,
    ! [X4: produc4862256710654508797_c_nat,Xs: list_P8155366108056616461_c_nat,Ys: list_P2602636364039601413_c_nat] :
      ( ( compos1451257573582057279_c_nat @ ( cons_P7928565014262433479_c_nat @ X4 @ Xs ) @ Ys )
      = ( case_o2058764249211701737_c_nat @ ( compos1451257573582057279_c_nat @ ( delete4993458036327213456_c_nat @ ( produc8816143915943518219_c_nat @ X4 ) @ Xs ) @ Ys )
        @ ^ [V: option7520157102916957007_c_nat] : ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( produc8816143915943518219_c_nat @ X4 ) @ V ) @ ( compos1451257573582057279_c_nat @ Xs @ Ys ) )
        @ ( map_of1110926660169950083_c_nat @ Ys @ ( produc2435110264752509257_c_nat @ X4 ) ) ) ) ).

% compose.simps(2)
thf(fact_1018_distinct__compose,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,Ys: list_P5405424656961669729_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( compos1751932118770894235_c_nat @ Xs @ Ys ) ) ) ) ).

% distinct_compose
thf(fact_1019_distinct__compose,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,Ys: list_P7599720138408045768_nat_c] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( compos4753030616485652194_nat_c @ Xs @ Ys ) ) ) ) ).

% distinct_compose
thf(fact_1020_distinct__compose,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,Ys: list_P6797689553898924723_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs ) )
     => ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( compos3540800493588629485_c_nat @ Xs @ Ys ) ) ) ) ).

% distinct_compose
thf(fact_1021_distinct__compose,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P4207096184057010108_c_nat] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( compos5151074131599017758_c_nat @ Xs @ Ys ) ) ) ) ).

% distinct_compose
thf(fact_1022_distinct__compose,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P267710562049004973od_c_c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( compose_b_c_c @ Xs @ Ys ) ) ) ) ).

% distinct_compose
thf(fact_1023_distinct__compose,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P5760860764591065870_c_nat] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs ) )
     => ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( compos1359844409926042736_c_nat @ Xs @ Ys ) ) ) ) ).

% distinct_compose
thf(fact_1024_distinct__compose,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,Ys: list_P2361265211252882099_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( compos6055679653055957741_c_nat @ Xs @ Ys ) ) ) ) ).

% distinct_compose
thf(fact_1025_distinct__compose,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,Ys: list_P486377699747300854_nat_c] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( compos2116506346773900048_nat_c @ Xs @ Ys ) ) ) ) ).

% distinct_compose
thf(fact_1026_distinct__compose,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,Ys: list_P2602636364039601413_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs ) )
     => ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( compos1451257573582057279_c_nat @ Xs @ Ys ) ) ) ) ).

% distinct_compose
thf(fact_1027_compose_Opsimps_I2_J,axiom,
    ! [X4: produc6689720525226317995_c_nat,Xs: list_P7417839048565863355_c_nat,Ys: list_P6797689553898924723_c_nat] :
      ( ( accp_P3525169532204392036_c_nat @ compos700111631785239794_c_nat @ ( produc3935417639385211231_c_nat @ ( cons_P7294201633519847029_c_nat @ X4 @ Xs ) @ Ys ) )
     => ( ( compos3540800493588629485_c_nat @ ( cons_P7294201633519847029_c_nat @ X4 @ Xs ) @ Ys )
        = ( case_o2058764249211701737_c_nat @ ( compos3540800493588629485_c_nat @ ( delete1755679665865935934_c_nat @ ( produc6033490199168946105_c_nat @ X4 ) @ Xs ) @ Ys )
          @ ^ [V: option7520157102916957007_c_nat] : ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( produc6033490199168946105_c_nat @ X4 ) @ V ) @ ( compos3540800493588629485_c_nat @ Xs @ Ys ) )
          @ ( map_of459223425620940849_c_nat @ Ys @ ( produc6319817161620345591_c_nat @ X4 ) ) ) ) ) ).

% compose.psimps(2)
thf(fact_1028_compose_Opsimps_I2_J,axiom,
    ! [X4: product_prod_b_c,Xs: list_P903359562653991662od_b_c,Ys: list_P5760860764591065870_c_nat] :
      ( ( accp_P8481634180717613666_c_nat @ compos3893318268482025771_c_nat @ ( produc8347930384216144549_c_nat @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs ) @ Ys ) )
     => ( ( compos1359844409926042736_c_nat @ ( cons_P4529483553340347422od_b_c @ X4 @ Xs ) @ Ys )
        = ( case_o2058764249211701737_c_nat @ ( compos1359844409926042736_c_nat @ ( delete_b_c @ ( product_fst_b_c @ X4 ) @ Xs ) @ Ys )
          @ ^ [V: option7520157102916957007_c_nat] : ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( product_fst_b_c @ X4 ) @ V ) @ ( compos1359844409926042736_c_nat @ Xs @ Ys ) )
          @ ( map_of5971217104590012204_c_nat @ Ys @ ( product_snd_b_c @ X4 ) ) ) ) ) ).

% compose.psimps(2)
thf(fact_1029_compose_Opsimps_I2_J,axiom,
    ! [X4: produc4862256710654508797_c_nat,Xs: list_P8155366108056616461_c_nat,Ys: list_P2602636364039601413_c_nat] :
      ( ( accp_P7820116637038163428_c_nat @ compos2150493326690376004_c_nat @ ( produc1079876897952166367_c_nat @ ( cons_P7928565014262433479_c_nat @ X4 @ Xs ) @ Ys ) )
     => ( ( compos1451257573582057279_c_nat @ ( cons_P7928565014262433479_c_nat @ X4 @ Xs ) @ Ys )
        = ( case_o2058764249211701737_c_nat @ ( compos1451257573582057279_c_nat @ ( delete4993458036327213456_c_nat @ ( produc8816143915943518219_c_nat @ X4 ) @ Xs ) @ Ys )
          @ ^ [V: option7520157102916957007_c_nat] : ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( produc8816143915943518219_c_nat @ X4 ) @ V ) @ ( compos1451257573582057279_c_nat @ Xs @ Ys ) )
          @ ( map_of1110926660169950083_c_nat @ Ys @ ( produc2435110264752509257_c_nat @ X4 ) ) ) ) ) ).

% compose.psimps(2)
thf(fact_1030_compose_Opelims,axiom,
    ! [X4: list_P7417839048565863355_c_nat,Xa: list_P6797689553898924723_c_nat,Y4: list_P8155366108056616461_c_nat] :
      ( ( ( compos3540800493588629485_c_nat @ X4 @ Xa )
        = Y4 )
     => ( ( accp_P3525169532204392036_c_nat @ compos700111631785239794_c_nat @ ( produc3935417639385211231_c_nat @ X4 @ Xa ) )
       => ( ( ( X4 = nil_Pr7000167559216756261_c_nat )
           => ( ( Y4 = nil_Pr2407244426195543671_c_nat )
             => ~ ( accp_P3525169532204392036_c_nat @ compos700111631785239794_c_nat @ ( produc3935417639385211231_c_nat @ nil_Pr7000167559216756261_c_nat @ Xa ) ) ) )
         => ~ ! [X: produc6689720525226317995_c_nat,Xs4: list_P7417839048565863355_c_nat] :
                ( ( X4
                  = ( cons_P7294201633519847029_c_nat @ X @ Xs4 ) )
               => ( ( Y4
                    = ( case_o2058764249211701737_c_nat @ ( compos3540800493588629485_c_nat @ ( delete1755679665865935934_c_nat @ ( produc6033490199168946105_c_nat @ X ) @ Xs4 ) @ Xa )
                      @ ^ [V: option7520157102916957007_c_nat] : ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( produc6033490199168946105_c_nat @ X ) @ V ) @ ( compos3540800493588629485_c_nat @ Xs4 @ Xa ) )
                      @ ( map_of459223425620940849_c_nat @ Xa @ ( produc6319817161620345591_c_nat @ X ) ) ) )
                 => ~ ( accp_P3525169532204392036_c_nat @ compos700111631785239794_c_nat @ ( produc3935417639385211231_c_nat @ ( cons_P7294201633519847029_c_nat @ X @ Xs4 ) @ Xa ) ) ) ) ) ) ) ).

% compose.pelims
thf(fact_1031_compose_Opelims,axiom,
    ! [X4: list_P903359562653991662od_b_c,Xa: list_P5760860764591065870_c_nat,Y4: list_P8155366108056616461_c_nat] :
      ( ( ( compos1359844409926042736_c_nat @ X4 @ Xa )
        = Y4 )
     => ( ( accp_P8481634180717613666_c_nat @ compos3893318268482025771_c_nat @ ( produc8347930384216144549_c_nat @ X4 @ Xa ) )
       => ( ( ( X4 = nil_Product_prod_b_c )
           => ( ( Y4 = nil_Pr2407244426195543671_c_nat )
             => ~ ( accp_P8481634180717613666_c_nat @ compos3893318268482025771_c_nat @ ( produc8347930384216144549_c_nat @ nil_Product_prod_b_c @ Xa ) ) ) )
         => ~ ! [X: product_prod_b_c,Xs4: list_P903359562653991662od_b_c] :
                ( ( X4
                  = ( cons_P4529483553340347422od_b_c @ X @ Xs4 ) )
               => ( ( Y4
                    = ( case_o2058764249211701737_c_nat @ ( compos1359844409926042736_c_nat @ ( delete_b_c @ ( product_fst_b_c @ X ) @ Xs4 ) @ Xa )
                      @ ^ [V: option7520157102916957007_c_nat] : ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( product_fst_b_c @ X ) @ V ) @ ( compos1359844409926042736_c_nat @ Xs4 @ Xa ) )
                      @ ( map_of5971217104590012204_c_nat @ Xa @ ( product_snd_b_c @ X ) ) ) )
                 => ~ ( accp_P8481634180717613666_c_nat @ compos3893318268482025771_c_nat @ ( produc8347930384216144549_c_nat @ ( cons_P4529483553340347422od_b_c @ X @ Xs4 ) @ Xa ) ) ) ) ) ) ) ).

% compose.pelims
thf(fact_1032_compose_Opelims,axiom,
    ! [X4: list_P8155366108056616461_c_nat,Xa: list_P2602636364039601413_c_nat,Y4: list_P8155366108056616461_c_nat] :
      ( ( ( compos1451257573582057279_c_nat @ X4 @ Xa )
        = Y4 )
     => ( ( accp_P7820116637038163428_c_nat @ compos2150493326690376004_c_nat @ ( produc1079876897952166367_c_nat @ X4 @ Xa ) )
       => ( ( ( X4 = nil_Pr2407244426195543671_c_nat )
           => ( ( Y4 = nil_Pr2407244426195543671_c_nat )
             => ~ ( accp_P7820116637038163428_c_nat @ compos2150493326690376004_c_nat @ ( produc1079876897952166367_c_nat @ nil_Pr2407244426195543671_c_nat @ Xa ) ) ) )
         => ~ ! [X: produc4862256710654508797_c_nat,Xs4: list_P8155366108056616461_c_nat] :
                ( ( X4
                  = ( cons_P7928565014262433479_c_nat @ X @ Xs4 ) )
               => ( ( Y4
                    = ( case_o2058764249211701737_c_nat @ ( compos1451257573582057279_c_nat @ ( delete4993458036327213456_c_nat @ ( produc8816143915943518219_c_nat @ X ) @ Xs4 ) @ Xa )
                      @ ^ [V: option7520157102916957007_c_nat] : ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ ( produc8816143915943518219_c_nat @ X ) @ V ) @ ( compos1451257573582057279_c_nat @ Xs4 @ Xa ) )
                      @ ( map_of1110926660169950083_c_nat @ Xa @ ( produc2435110264752509257_c_nat @ X ) ) ) )
                 => ~ ( accp_P7820116637038163428_c_nat @ compos2150493326690376004_c_nat @ ( produc1079876897952166367_c_nat @ ( cons_P7928565014262433479_c_nat @ X @ Xs4 ) @ Xa ) ) ) ) ) ) ) ).

% compose.pelims
thf(fact_1033_map__append,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,Ys: list_P7417839048565863355_c_nat] :
      ( ( map_Pr6200325787298368846_nat_b @ F @ ( append7611157511503221456_c_nat @ Xs @ Ys ) )
      = ( append_b @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) @ ( map_Pr6200325787298368846_nat_b @ F @ Ys ) ) ) ).

% map_append
thf(fact_1034_map__append,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( map_Pr8222292247188198875_b_c_b @ F @ ( append2547753245680614915od_b_c @ Xs @ Ys ) )
      = ( append_b @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) @ ( map_Pr8222292247188198875_b_c_b @ F @ Ys ) ) ) ).

% map_append
thf(fact_1035_map__eq__append__conv,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,Ys: list_b,Zs: list_b] :
      ( ( ( map_Pr6200325787298368846_nat_b @ F @ Xs )
        = ( append_b @ Ys @ Zs ) )
      = ( ? [Us: list_P7417839048565863355_c_nat,Vs: list_P7417839048565863355_c_nat] :
            ( ( Xs
              = ( append7611157511503221456_c_nat @ Us @ Vs ) )
            & ( Ys
              = ( map_Pr6200325787298368846_nat_b @ F @ Us ) )
            & ( Zs
              = ( map_Pr6200325787298368846_nat_b @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_1036_map__eq__append__conv,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,Ys: list_b,Zs: list_b] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
        = ( append_b @ Ys @ Zs ) )
      = ( ? [Us: list_P903359562653991662od_b_c,Vs: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( append2547753245680614915od_b_c @ Us @ Vs ) )
            & ( Ys
              = ( map_Pr8222292247188198875_b_c_b @ F @ Us ) )
            & ( Zs
              = ( map_Pr8222292247188198875_b_c_b @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_1037_append__eq__map__conv,axiom,
    ! [Ys: list_b,Zs: list_b,F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat] :
      ( ( ( append_b @ Ys @ Zs )
        = ( map_Pr6200325787298368846_nat_b @ F @ Xs ) )
      = ( ? [Us: list_P7417839048565863355_c_nat,Vs: list_P7417839048565863355_c_nat] :
            ( ( Xs
              = ( append7611157511503221456_c_nat @ Us @ Vs ) )
            & ( Ys
              = ( map_Pr6200325787298368846_nat_b @ F @ Us ) )
            & ( Zs
              = ( map_Pr6200325787298368846_nat_b @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_1038_append__eq__map__conv,axiom,
    ! [Ys: list_b,Zs: list_b,F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( ( append_b @ Ys @ Zs )
        = ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) )
      = ( ? [Us: list_P903359562653991662od_b_c,Vs: list_P903359562653991662od_b_c] :
            ( ( Xs
              = ( append2547753245680614915od_b_c @ Us @ Vs ) )
            & ( Ys
              = ( map_Pr8222292247188198875_b_c_b @ F @ Us ) )
            & ( Zs
              = ( map_Pr8222292247188198875_b_c_b @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_1039_product_Osimps_I2_J,axiom,
    ! [X4: b,Xs: list_b,Ys: list_o9079336047566018143_c_nat] :
      ( ( produc5715903733481046217_c_nat @ ( cons_b @ X4 @ Xs ) @ Ys )
      = ( append4735040753520518690_c_nat @ ( map_op2222718326440564242_c_nat @ ( produc5716802255202478839_c_nat @ X4 ) @ Ys ) @ ( produc5715903733481046217_c_nat @ Xs @ Ys ) ) ) ).

% product.simps(2)
thf(fact_1040_product_Osimps_I2_J,axiom,
    ! [X4: c,Xs: list_c,Ys: list_nat] :
      ( ( product_c_nat @ ( cons_c @ X4 @ Xs ) @ Ys )
      = ( append926756200616138276_c_nat @ ( map_na7150870762157906453_c_nat @ ( product_Pair_c_nat @ X4 ) @ Ys ) @ ( product_c_nat @ Xs @ Ys ) ) ) ).

% product.simps(2)
thf(fact_1041_dom__update,axiom,
    ! [K2: b,V4: produc4862256710654508797_c_nat,Al: list_P7417839048565863355_c_nat] :
      ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ ( update1129433653605194368_c_nat @ K2 @ V4 @ Al ) ) )
      = ( sup_sup_set_b @ ( insert_b @ K2 @ bot_bot_set_b ) @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Al ) ) ) ) ).

% dom_update
thf(fact_1042_dom__update,axiom,
    ! [K2: b,V4: c,Al: list_P903359562653991662od_b_c] :
      ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( update_b_c @ K2 @ V4 @ Al ) ) )
      = ( sup_sup_set_b @ ( insert_b @ K2 @ bot_bot_set_b ) @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Al ) ) ) ) ).

% dom_update
thf(fact_1043_dom__update,axiom,
    ! [K2: b,V4: option7520157102916957007_c_nat,Al: list_P8155366108056616461_c_nat] :
      ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ ( update7891854496680085970_c_nat @ K2 @ V4 @ Al ) ) )
      = ( sup_sup_set_b @ ( insert_b @ K2 @ bot_bot_set_b ) @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Al ) ) ) ) ).

% dom_update
thf(fact_1044_update__keys,axiom,
    ! [K2: b,Al: list_P7417839048565863355_c_nat,V4: produc4862256710654508797_c_nat] :
      ( ( ( member_b @ K2 @ ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) ) )
       => ( ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( update1129433653605194368_c_nat @ K2 @ V4 @ Al ) )
          = ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) ) )
      & ( ~ ( member_b @ K2 @ ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) ) )
       => ( ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( update1129433653605194368_c_nat @ K2 @ V4 @ Al ) )
          = ( append_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) @ ( cons_b @ K2 @ nil_b ) ) ) ) ) ).

% update_keys
thf(fact_1045_update__keys,axiom,
    ! [K2: b,Al: list_P903359562653991662od_b_c,V4: c] :
      ( ( ( member_b @ K2 @ ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) ) )
       => ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( update_b_c @ K2 @ V4 @ Al ) )
          = ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) ) )
      & ( ~ ( member_b @ K2 @ ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) ) )
       => ( ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( update_b_c @ K2 @ V4 @ Al ) )
          = ( append_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) @ ( cons_b @ K2 @ nil_b ) ) ) ) ) ).

% update_keys
thf(fact_1046_update__keys,axiom,
    ! [K2: b,Al: list_P8155366108056616461_c_nat,V4: option7520157102916957007_c_nat] :
      ( ( ( member_b @ K2 @ ( set_b2 @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) ) )
       => ( ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( update7891854496680085970_c_nat @ K2 @ V4 @ Al ) )
          = ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) ) )
      & ( ~ ( member_b @ K2 @ ( set_b2 @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) ) )
       => ( ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( update7891854496680085970_c_nat @ K2 @ V4 @ Al ) )
          = ( append_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) @ ( cons_b @ K2 @ nil_b ) ) ) ) ) ).

% update_keys
thf(fact_1047_mmap__update__def,axiom,
    mmap_u2662500248198004552_c_nat = update1129433653605194368_c_nat ).

% mmap_update_def
thf(fact_1048_mmap__update__def,axiom,
    mmap_update_b_c = update_b_c ).

% mmap_update_def
thf(fact_1049_update__filter,axiom,
    ! [A: b,K2: b,V4: produc4862256710654508797_c_nat,Ps: list_P7417839048565863355_c_nat] :
      ( ( A != K2 )
     => ( ( update1129433653605194368_c_nat @ K2 @ V4
          @ ( filter3549032352921181294_c_nat
            @ ^ [Q5: produc6689720525226317995_c_nat] :
                ( ( produc6033490199168946105_c_nat @ Q5 )
               != A )
            @ Ps ) )
        = ( filter3549032352921181294_c_nat
          @ ^ [Q5: produc6689720525226317995_c_nat] :
              ( ( produc6033490199168946105_c_nat @ Q5 )
             != A )
          @ ( update1129433653605194368_c_nat @ K2 @ V4 @ Ps ) ) ) ) ).

% update_filter
thf(fact_1050_update__filter,axiom,
    ! [A: b,K2: b,V4: c,Ps: list_P903359562653991662od_b_c] :
      ( ( A != K2 )
     => ( ( update_b_c @ K2 @ V4
          @ ( filter4794301956807392997od_b_c
            @ ^ [Q5: product_prod_b_c] :
                ( ( product_fst_b_c @ Q5 )
               != A )
            @ Ps ) )
        = ( filter4794301956807392997od_b_c
          @ ^ [Q5: product_prod_b_c] :
              ( ( product_fst_b_c @ Q5 )
             != A )
          @ ( update_b_c @ K2 @ V4 @ Ps ) ) ) ) ).

% update_filter
thf(fact_1051_update__filter,axiom,
    ! [A: b,K2: b,V4: option7520157102916957007_c_nat,Ps: list_P8155366108056616461_c_nat] :
      ( ( A != K2 )
     => ( ( update7891854496680085970_c_nat @ K2 @ V4
          @ ( filter4765930412304252352_c_nat
            @ ^ [Q5: produc4862256710654508797_c_nat] :
                ( ( produc8816143915943518219_c_nat @ Q5 )
               != A )
            @ Ps ) )
        = ( filter4765930412304252352_c_nat
          @ ^ [Q5: produc4862256710654508797_c_nat] :
              ( ( produc8816143915943518219_c_nat @ Q5 )
             != A )
          @ ( update7891854496680085970_c_nat @ K2 @ V4 @ Ps ) ) ) ) ).

% update_filter
thf(fact_1052_update_Osimps_I2_J,axiom,
    ! [P: product_prod_c_nat,K2: c,V4: nat,Ps: list_P5561950507774946575_c_nat] :
      ( ( ( ( product_fst_c_nat @ P )
          = K2 )
       => ( ( update_c_nat @ K2 @ V4 @ ( cons_P7676055461599297353_c_nat @ P @ Ps ) )
          = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K2 @ V4 ) @ Ps ) ) )
      & ( ( ( product_fst_c_nat @ P )
         != K2 )
       => ( ( update_c_nat @ K2 @ V4 @ ( cons_P7676055461599297353_c_nat @ P @ Ps ) )
          = ( cons_P7676055461599297353_c_nat @ P @ ( update_c_nat @ K2 @ V4 @ Ps ) ) ) ) ) ).

% update.simps(2)
thf(fact_1053_update_Osimps_I2_J,axiom,
    ! [P: produc6689720525226317995_c_nat,K2: b,V4: produc4862256710654508797_c_nat,Ps: list_P7417839048565863355_c_nat] :
      ( ( ( ( produc6033490199168946105_c_nat @ P )
          = K2 )
       => ( ( update1129433653605194368_c_nat @ K2 @ V4 @ ( cons_P7294201633519847029_c_nat @ P @ Ps ) )
          = ( cons_P7294201633519847029_c_nat @ ( produc8868828497325608613_c_nat @ K2 @ V4 ) @ Ps ) ) )
      & ( ( ( produc6033490199168946105_c_nat @ P )
         != K2 )
       => ( ( update1129433653605194368_c_nat @ K2 @ V4 @ ( cons_P7294201633519847029_c_nat @ P @ Ps ) )
          = ( cons_P7294201633519847029_c_nat @ P @ ( update1129433653605194368_c_nat @ K2 @ V4 @ Ps ) ) ) ) ) ).

% update.simps(2)
thf(fact_1054_update_Osimps_I2_J,axiom,
    ! [P: product_prod_b_c,K2: b,V4: c,Ps: list_P903359562653991662od_b_c] :
      ( ( ( ( product_fst_b_c @ P )
          = K2 )
       => ( ( update_b_c @ K2 @ V4 @ ( cons_P4529483553340347422od_b_c @ P @ Ps ) )
          = ( cons_P4529483553340347422od_b_c @ ( product_Pair_b_c @ K2 @ V4 ) @ Ps ) ) )
      & ( ( ( product_fst_b_c @ P )
         != K2 )
       => ( ( update_b_c @ K2 @ V4 @ ( cons_P4529483553340347422od_b_c @ P @ Ps ) )
          = ( cons_P4529483553340347422od_b_c @ P @ ( update_b_c @ K2 @ V4 @ Ps ) ) ) ) ) ).

% update.simps(2)
thf(fact_1055_update_Osimps_I2_J,axiom,
    ! [P: produc4862256710654508797_c_nat,K2: b,V4: option7520157102916957007_c_nat,Ps: list_P8155366108056616461_c_nat] :
      ( ( ( ( produc8816143915943518219_c_nat @ P )
          = K2 )
       => ( ( update7891854496680085970_c_nat @ K2 @ V4 @ ( cons_P7928565014262433479_c_nat @ P @ Ps ) )
          = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ V4 ) @ Ps ) ) )
      & ( ( ( produc8816143915943518219_c_nat @ P )
         != K2 )
       => ( ( update7891854496680085970_c_nat @ K2 @ V4 @ ( cons_P7928565014262433479_c_nat @ P @ Ps ) )
          = ( cons_P7928565014262433479_c_nat @ P @ ( update7891854496680085970_c_nat @ K2 @ V4 @ Ps ) ) ) ) ) ).

% update.simps(2)
thf(fact_1056_distinct__update,axiom,
    ! [Al: list_P7417839048565863355_c_nat,K2: b,V4: produc4862256710654508797_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( update1129433653605194368_c_nat @ K2 @ V4 @ Al ) ) ) ) ).

% distinct_update
thf(fact_1057_distinct__update,axiom,
    ! [Al: list_P903359562653991662od_b_c,K2: b,V4: c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( update_b_c @ K2 @ V4 @ Al ) ) ) ) ).

% distinct_update
thf(fact_1058_distinct__update,axiom,
    ! [Al: list_P8155366108056616461_c_nat,K2: b,V4: option7520157102916957007_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) )
     => ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( update7891854496680085970_c_nat @ K2 @ V4 @ Al ) ) ) ) ).

% distinct_update
thf(fact_1059_update_Osimps_I1_J,axiom,
    ! [K2: b,V4: option7520157102916957007_c_nat] :
      ( ( update7891854496680085970_c_nat @ K2 @ V4 @ nil_Pr2407244426195543671_c_nat )
      = ( cons_P7928565014262433479_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ V4 ) @ nil_Pr2407244426195543671_c_nat ) ) ).

% update.simps(1)
thf(fact_1060_update_Osimps_I1_J,axiom,
    ! [K2: c,V4: nat] :
      ( ( update_c_nat @ K2 @ V4 @ nil_Pr650041864559007481_c_nat )
      = ( cons_P7676055461599297353_c_nat @ ( product_Pair_c_nat @ K2 @ V4 ) @ nil_Pr650041864559007481_c_nat ) ) ).

% update.simps(1)
thf(fact_1061_distinct__clearjunk__id,axiom,
    ! [Al: list_P7417839048565863355_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) )
     => ( ( clearj6941515884458365402_c_nat @ Al )
        = Al ) ) ).

% distinct_clearjunk_id
thf(fact_1062_distinct__clearjunk__id,axiom,
    ! [Al: list_P903359562653991662od_b_c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) )
     => ( ( clearjunk_b_c @ Al )
        = Al ) ) ).

% distinct_clearjunk_id
thf(fact_1063_distinct__clearjunk__id,axiom,
    ! [Al: list_P8155366108056616461_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) )
     => ( ( clearj1988084348930812716_c_nat @ Al )
        = Al ) ) ).

% distinct_clearjunk_id
thf(fact_1064_clearjunk__map,axiom,
    ! [F: produc6689720525226317995_c_nat > produc6689720525226317995_c_nat,Ps: list_P7417839048565863355_c_nat] :
      ( ! [Kv: produc6689720525226317995_c_nat] :
          ( ( produc6033490199168946105_c_nat @ ( F @ Kv ) )
          = ( produc6033490199168946105_c_nat @ Kv ) )
     => ( ( clearj6941515884458365402_c_nat @ ( map_Pr7976378254695727132_c_nat @ F @ Ps ) )
        = ( map_Pr7976378254695727132_c_nat @ F @ ( clearj6941515884458365402_c_nat @ Ps ) ) ) ) ).

% clearjunk_map
thf(fact_1065_clearjunk__map,axiom,
    ! [F: product_prod_b_c > produc6689720525226317995_c_nat,Ps: list_P903359562653991662od_b_c] :
      ( ! [Kv: product_prod_b_c] :
          ( ( produc6033490199168946105_c_nat @ ( F @ Kv ) )
          = ( product_fst_b_c @ Kv ) )
     => ( ( clearj6941515884458365402_c_nat @ ( map_Pr6155953585549333711_c_nat @ F @ Ps ) )
        = ( map_Pr6155953585549333711_c_nat @ F @ ( clearjunk_b_c @ Ps ) ) ) ) ).

% clearjunk_map
thf(fact_1066_clearjunk__map,axiom,
    ! [F: produc4862256710654508797_c_nat > produc6689720525226317995_c_nat,Ps: list_P8155366108056616461_c_nat] :
      ( ! [Kv: produc4862256710654508797_c_nat] :
          ( ( produc6033490199168946105_c_nat @ ( F @ Kv ) )
          = ( produc8816143915943518219_c_nat @ Kv ) )
     => ( ( clearj6941515884458365402_c_nat @ ( map_Pr5604758227450724974_c_nat @ F @ Ps ) )
        = ( map_Pr5604758227450724974_c_nat @ F @ ( clearj1988084348930812716_c_nat @ Ps ) ) ) ) ).

% clearjunk_map
thf(fact_1067_clearjunk__map,axiom,
    ! [F: produc6689720525226317995_c_nat > product_prod_b_c,Ps: list_P7417839048565863355_c_nat] :
      ( ! [Kv: produc6689720525226317995_c_nat] :
          ( ( product_fst_b_c @ ( F @ Kv ) )
          = ( produc6033490199168946105_c_nat @ Kv ) )
     => ( ( clearjunk_b_c @ ( map_Pr5949920527225541047od_b_c @ F @ Ps ) )
        = ( map_Pr5949920527225541047od_b_c @ F @ ( clearj6941515884458365402_c_nat @ Ps ) ) ) ) ).

% clearjunk_map
thf(fact_1068_clearjunk__map,axiom,
    ! [F: product_prod_b_c > product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
      ( ! [Kv: product_prod_b_c] :
          ( ( product_fst_b_c @ ( F @ Kv ) )
          = ( product_fst_b_c @ Kv ) )
     => ( ( clearjunk_b_c @ ( map_Pr6384634858812015940od_b_c @ F @ Ps ) )
        = ( map_Pr6384634858812015940od_b_c @ F @ ( clearjunk_b_c @ Ps ) ) ) ) ).

% clearjunk_map
thf(fact_1069_clearjunk__map,axiom,
    ! [F: produc4862256710654508797_c_nat > product_prod_b_c,Ps: list_P8155366108056616461_c_nat] :
      ( ! [Kv: produc4862256710654508797_c_nat] :
          ( ( product_fst_b_c @ ( F @ Kv ) )
          = ( produc8816143915943518219_c_nat @ Kv ) )
     => ( ( clearjunk_b_c @ ( map_Pr8130720921054730469od_b_c @ F @ Ps ) )
        = ( map_Pr8130720921054730469od_b_c @ F @ ( clearj1988084348930812716_c_nat @ Ps ) ) ) ) ).

% clearjunk_map
thf(fact_1070_clearjunk__map,axiom,
    ! [F: produc6689720525226317995_c_nat > produc4862256710654508797_c_nat,Ps: list_P7417839048565863355_c_nat] :
      ( ! [Kv: produc6689720525226317995_c_nat] :
          ( ( produc8816143915943518219_c_nat @ ( F @ Kv ) )
          = ( produc6033490199168946105_c_nat @ Kv ) )
     => ( ( clearj1988084348930812716_c_nat @ ( map_Pr4595408008933130094_c_nat @ F @ Ps ) )
        = ( map_Pr4595408008933130094_c_nat @ F @ ( clearj6941515884458365402_c_nat @ Ps ) ) ) ) ).

% clearjunk_map
thf(fact_1071_clearjunk__map,axiom,
    ! [F: product_prod_b_c > produc4862256710654508797_c_nat,Ps: list_P903359562653991662od_b_c] :
      ( ! [Kv: product_prod_b_c] :
          ( ( produc8816143915943518219_c_nat @ ( F @ Kv ) )
          = ( product_fst_b_c @ Kv ) )
     => ( ( clearj1988084348930812716_c_nat @ ( map_Pr2101552211129450273_c_nat @ F @ Ps ) )
        = ( map_Pr2101552211129450273_c_nat @ F @ ( clearjunk_b_c @ Ps ) ) ) ) ).

% clearjunk_map
thf(fact_1072_clearjunk__map,axiom,
    ! [F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,Ps: list_P8155366108056616461_c_nat] :
      ( ! [Kv: produc4862256710654508797_c_nat] :
          ( ( produc8816143915943518219_c_nat @ ( F @ Kv ) )
          = ( produc8816143915943518219_c_nat @ Kv ) )
     => ( ( clearj1988084348930812716_c_nat @ ( map_Pr5967270692098401216_c_nat @ F @ Ps ) )
        = ( map_Pr5967270692098401216_c_nat @ F @ ( clearj1988084348930812716_c_nat @ Ps ) ) ) ) ).

% clearjunk_map
thf(fact_1073_clearjunk__keys__set,axiom,
    ! [Al: list_P7417839048565863355_c_nat] :
      ( ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( clearj6941515884458365402_c_nat @ Al ) ) )
      = ( set_b2 @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) ) ) ).

% clearjunk_keys_set
thf(fact_1074_clearjunk__keys__set,axiom,
    ! [Al: list_P903359562653991662od_b_c] :
      ( ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( clearjunk_b_c @ Al ) ) )
      = ( set_b2 @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) ) ) ).

% clearjunk_keys_set
thf(fact_1075_clearjunk__keys__set,axiom,
    ! [Al: list_P8155366108056616461_c_nat] :
      ( ( set_b2 @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( clearj1988084348930812716_c_nat @ Al ) ) )
      = ( set_b2 @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) ) ) ).

% clearjunk_keys_set
thf(fact_1076_distinct__clearjunk,axiom,
    ! [Al: list_P7417839048565863355_c_nat] : ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( clearj6941515884458365402_c_nat @ Al ) ) ) ).

% distinct_clearjunk
thf(fact_1077_distinct__clearjunk,axiom,
    ! [Al: list_P903359562653991662od_b_c] : ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( clearjunk_b_c @ Al ) ) ) ).

% distinct_clearjunk
thf(fact_1078_distinct__clearjunk,axiom,
    ! [Al: list_P8155366108056616461_c_nat] : ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( clearj1988084348930812716_c_nat @ Al ) ) ) ).

% distinct_clearjunk
thf(fact_1079_dom__clearjunk,axiom,
    ! [Al: list_P7417839048565863355_c_nat] :
      ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ ( clearj6941515884458365402_c_nat @ Al ) ) )
      = ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Al ) ) ) ).

% dom_clearjunk
thf(fact_1080_dom__clearjunk,axiom,
    ! [Al: list_P903359562653991662od_b_c] :
      ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( clearjunk_b_c @ Al ) ) )
      = ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Al ) ) ) ).

% dom_clearjunk
thf(fact_1081_dom__clearjunk,axiom,
    ! [Al: list_P8155366108056616461_c_nat] :
      ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ ( clearj1988084348930812716_c_nat @ Al ) ) )
      = ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Al ) ) ) ).

% dom_clearjunk
thf(fact_1082_distinct__updates,axiom,
    ! [Al: list_P7417839048565863355_c_nat,Ks: list_b,Vs2: list_P8155366108056616461_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( update9219691110958068421_c_nat @ Ks @ Vs2 @ Al ) ) ) ) ).

% distinct_updates
thf(fact_1083_distinct__updates,axiom,
    ! [Al: list_P903359562653991662od_b_c,Ks: list_b,Vs2: list_c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( updates_b_c @ Ks @ Vs2 @ Al ) ) ) ) ).

% distinct_updates
thf(fact_1084_distinct__updates,axiom,
    ! [Al: list_P8155366108056616461_c_nat,Ks: list_b,Vs2: list_o9079336047566018143_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) )
     => ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( update1185360923125034775_c_nat @ Ks @ Vs2 @ Al ) ) ) ) ).

% distinct_updates
thf(fact_1085_map__of__mapk__SomeI,axiom,
    ! [F: b > b,T: list_P8155366108056616461_c_nat,K2: b,X4: option7520157102916957007_c_nat] :
      ( ( inj_on_b_b @ F @ top_top_set_b )
     => ( ( ( map_of2598500733885366699_c_nat @ T @ K2 )
          = ( some_o7552056839342673834_c_nat @ X4 ) )
       => ( ( map_of2598500733885366699_c_nat
            @ ( map_Pr5967270692098401216_c_nat
              @ ( produc3722540595886809633_c_nat
                @ ^ [K: b] : ( produc5716802255202478839_c_nat @ ( F @ K ) ) )
              @ T )
            @ ( F @ K2 ) )
          = ( some_o7552056839342673834_c_nat @ X4 ) ) ) ) ).

% map_of_mapk_SomeI
thf(fact_1086_Id__onI,axiom,
    ! [A: b,A4: set_b] :
      ( ( member_b @ A @ A4 )
     => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ A ) @ ( id_on_b @ A4 ) ) ) ).

% Id_onI
thf(fact_1087_inj__map,axiom,
    ! [F: produc6689720525226317995_c_nat > b] :
      ( ( inj_on6407831275025639511list_b @ ( map_Pr6200325787298368846_nat_b @ F ) @ top_to7870874601094680993_c_nat )
      = ( inj_on1282781620109521761_nat_b @ F @ top_to689330392550168593_c_nat ) ) ).

% inj_map
thf(fact_1088_inj__map,axiom,
    ! [F: product_prod_b_c > b] :
      ( ( inj_on5485488043073140360list_b @ ( map_Pr8222292247188198875_b_c_b @ F ) @ top_to1393949755257851422od_b_c )
      = ( inj_on80528130674425864_b_c_b @ F @ top_to7569790511745352472od_b_c ) ) ).

% inj_map
thf(fact_1089_inj__mapI,axiom,
    ! [F: produc6689720525226317995_c_nat > b] :
      ( ( inj_on1282781620109521761_nat_b @ F @ top_to689330392550168593_c_nat )
     => ( inj_on6407831275025639511list_b @ ( map_Pr6200325787298368846_nat_b @ F ) @ top_to7870874601094680993_c_nat ) ) ).

% inj_mapI
thf(fact_1090_inj__mapI,axiom,
    ! [F: product_prod_b_c > b] :
      ( ( inj_on80528130674425864_b_c_b @ F @ top_to7569790511745352472od_b_c )
     => ( inj_on5485488043073140360list_b @ ( map_Pr8222292247188198875_b_c_b @ F ) @ top_to1393949755257851422od_b_c ) ) ).

% inj_mapI
thf(fact_1091_inj__map__eq__map,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,Ys: list_P7417839048565863355_c_nat] :
      ( ( inj_on1282781620109521761_nat_b @ F @ top_to689330392550168593_c_nat )
     => ( ( ( map_Pr6200325787298368846_nat_b @ F @ Xs )
          = ( map_Pr6200325787298368846_nat_b @ F @ Ys ) )
        = ( Xs = Ys ) ) ) ).

% inj_map_eq_map
thf(fact_1092_inj__map__eq__map,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( inj_on80528130674425864_b_c_b @ F @ top_to7569790511745352472od_b_c )
     => ( ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
          = ( map_Pr8222292247188198875_b_c_b @ F @ Ys ) )
        = ( Xs = Ys ) ) ) ).

% inj_map_eq_map
thf(fact_1093_FieldI2,axiom,
    ! [I2: b,J2: b,R: set_Product_prod_b_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ I2 @ J2 ) @ R )
     => ( member_b @ J2 @ ( field_b @ R ) ) ) ).

% FieldI2
thf(fact_1094_FieldI1,axiom,
    ! [I2: b,J2: b,R: set_Product_prod_b_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ I2 @ J2 ) @ R )
     => ( member_b @ I2 @ ( field_b @ R ) ) ) ).

% FieldI1
thf(fact_1095_inj__on__convol__ident,axiom,
    ! [F: b > option7520157102916957007_c_nat,X7: set_b] :
      ( inj_on5234885906491436875_c_nat
      @ ^ [X3: b] : ( produc5716802255202478839_c_nat @ X3 @ ( F @ X3 ) )
      @ X7 ) ).

% inj_on_convol_ident
thf(fact_1096_inj__on__convol__ident,axiom,
    ! [F: c > nat,X7: set_c] :
      ( inj_on2653622697545954510_c_nat
      @ ^ [X3: c] : ( product_Pair_c_nat @ X3 @ ( F @ X3 ) )
      @ X7 ) ).

% inj_on_convol_ident
thf(fact_1097_Id__onE,axiom,
    ! [C: product_prod_b_b,A4: set_b] :
      ( ( member7862447936710763792od_b_b @ C @ ( id_on_b @ A4 ) )
     => ~ ! [X: b] :
            ( ( member_b @ X @ A4 )
           => ( C
             != ( product_Pair_b_b @ X @ X ) ) ) ) ).

% Id_onE
thf(fact_1098_Id__on__eqI,axiom,
    ! [A: b,B: b,A4: set_b] :
      ( ( A = B )
     => ( ( member_b @ A @ A4 )
       => ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( id_on_b @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_1099_Id__on__iff,axiom,
    ! [X4: b,Y4: b,A4: set_b] :
      ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ ( id_on_b @ A4 ) )
      = ( ( X4 = Y4 )
        & ( member_b @ X4 @ A4 ) ) ) ).

% Id_on_iff
thf(fact_1100_inj__mapD,axiom,
    ! [F: produc6689720525226317995_c_nat > b] :
      ( ( inj_on6407831275025639511list_b @ ( map_Pr6200325787298368846_nat_b @ F ) @ top_to7870874601094680993_c_nat )
     => ( inj_on1282781620109521761_nat_b @ F @ top_to689330392550168593_c_nat ) ) ).

% inj_mapD
thf(fact_1101_inj__mapD,axiom,
    ! [F: product_prod_b_c > b] :
      ( ( inj_on5485488043073140360list_b @ ( map_Pr8222292247188198875_b_c_b @ F ) @ top_to1393949755257851422od_b_c )
     => ( inj_on80528130674425864_b_c_b @ F @ top_to7569790511745352472od_b_c ) ) ).

% inj_mapD
thf(fact_1102_inj__image__mem__iff,axiom,
    ! [F: b > b,A: b,A4: set_b] :
      ( ( inj_on_b_b @ F @ top_top_set_b )
     => ( ( member_b @ ( F @ A ) @ ( image_b_b @ F @ A4 ) )
        = ( member_b @ A @ A4 ) ) ) ).

% inj_image_mem_iff
thf(fact_1103_inj__img__insertE,axiom,
    ! [F: b > b,A4: set_b,X4: b,B5: set_b] :
      ( ( inj_on_b_b @ F @ A4 )
     => ( ~ ( member_b @ X4 @ B5 )
       => ( ( ( insert_b @ X4 @ B5 )
            = ( image_b_b @ F @ A4 ) )
         => ~ ! [X8: b,A7: set_b] :
                ( ~ ( member_b @ X8 @ A7 )
               => ( ( A4
                    = ( insert_b @ X8 @ A7 ) )
                 => ( ( X4
                      = ( F @ X8 ) )
                   => ( B5
                     != ( image_b_b @ F @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1104_map__injective,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,Ys: list_P7417839048565863355_c_nat] :
      ( ( ( map_Pr6200325787298368846_nat_b @ F @ Xs )
        = ( map_Pr6200325787298368846_nat_b @ F @ Ys ) )
     => ( ( inj_on1282781620109521761_nat_b @ F @ top_to689330392550168593_c_nat )
       => ( Xs = Ys ) ) ) ).

% map_injective
thf(fact_1105_map__injective,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
        = ( map_Pr8222292247188198875_b_c_b @ F @ Ys ) )
     => ( ( inj_on80528130674425864_b_c_b @ F @ top_to7569790511745352472od_b_c )
       => ( Xs = Ys ) ) ) ).

% map_injective
thf(fact_1106_swap__inj__on,axiom,
    ! [A4: set_Pr1066409585067842203_nat_b] :
      ( inj_on7414042157139895125_c_nat
      @ ( produc2411843245718984257_c_nat
        @ ^ [I: option7520157102916957007_c_nat,J: b] : ( produc5716802255202478839_c_nat @ J @ I ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_1107_swap__inj__on,axiom,
    ! [A4: set_Pr4335409928928841947_nat_c] :
      ( inj_on3988094872618758807_c_nat
      @ ( produc8194310905338674051_c_nat
        @ ^ [I: nat,J: c] : ( product_Pair_c_nat @ J @ I ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_1108_map__inj__on,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,Ys: list_P7417839048565863355_c_nat] :
      ( ( ( map_Pr6200325787298368846_nat_b @ F @ Xs )
        = ( map_Pr6200325787298368846_nat_b @ F @ Ys ) )
     => ( ( inj_on1282781620109521761_nat_b @ F @ ( sup_su2972426202601250957_c_nat @ ( set_Pr5807476769692288598_c_nat @ Xs ) @ ( set_Pr5807476769692288598_c_nat @ Ys ) ) )
       => ( Xs = Ys ) ) ) ).

% map_inj_on
thf(fact_1109_map__inj__on,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
        = ( map_Pr8222292247188198875_b_c_b @ F @ Ys ) )
     => ( ( inj_on80528130674425864_b_c_b @ F @ ( sup_su2554677861087362972od_b_c @ ( set_Product_prod_b_c2 @ Xs ) @ ( set_Product_prod_b_c2 @ Ys ) ) )
       => ( Xs = Ys ) ) ) ).

% map_inj_on
thf(fact_1110_inj__on__map__eq__map,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat,Ys: list_P7417839048565863355_c_nat] :
      ( ( inj_on1282781620109521761_nat_b @ F @ ( sup_su2972426202601250957_c_nat @ ( set_Pr5807476769692288598_c_nat @ Xs ) @ ( set_Pr5807476769692288598_c_nat @ Ys ) ) )
     => ( ( ( map_Pr6200325787298368846_nat_b @ F @ Xs )
          = ( map_Pr6200325787298368846_nat_b @ F @ Ys ) )
        = ( Xs = Ys ) ) ) ).

% inj_on_map_eq_map
thf(fact_1111_inj__on__map__eq__map,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( inj_on80528130674425864_b_c_b @ F @ ( sup_su2554677861087362972od_b_c @ ( set_Product_prod_b_c2 @ Xs ) @ ( set_Product_prod_b_c2 @ Ys ) ) )
     => ( ( ( map_Pr8222292247188198875_b_c_b @ F @ Xs )
          = ( map_Pr8222292247188198875_b_c_b @ F @ Ys ) )
        = ( Xs = Ys ) ) ) ).

% inj_on_map_eq_map
thf(fact_1112_distinct__map,axiom,
    ! [F: b > b,Xs: list_b] :
      ( ( distinct_b @ ( map_b_b @ F @ Xs ) )
      = ( ( distinct_b @ Xs )
        & ( inj_on_b_b @ F @ ( set_b2 @ Xs ) ) ) ) ).

% distinct_map
thf(fact_1113_distinct__map,axiom,
    ! [F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) )
      = ( ( distin8649924140525960306_c_nat @ Xs )
        & ( inj_on1282781620109521761_nat_b @ F @ ( set_Pr5807476769692288598_c_nat @ Xs ) ) ) ) ).

% distinct_map
thf(fact_1114_distinct__map,axiom,
    ! [F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) )
      = ( ( distin6568250333530618337od_b_c @ Xs )
        & ( inj_on80528130674425864_b_c_b @ F @ ( set_Product_prod_b_c2 @ Xs ) ) ) ) ).

% distinct_map
thf(fact_1115_cstep__def,axiom,
    ( cstep_4633390515900757979_c_nat
    = ( ^ [Step2: b > option7520157102916957007_c_nat > b,St3: mappin7688773591249790157_nat_b,Q5: b,Bs3: option7520157102916957007_c_nat] :
          ( case_o6950705069787775509at_b_b @ ( produc8656096559574149925_nat_b @ ( Step2 @ Q5 @ Bs3 ) @ ( update3096138758973351061_nat_b @ ( produc5716802255202478839_c_nat @ Q5 @ Bs3 ) @ ( Step2 @ Q5 @ Bs3 ) @ St3 ) )
          @ ^ [V: b] : ( produc8656096559574149925_nat_b @ V @ St3 )
          @ ( lookup5460638230741284100_nat_b @ St3 @ ( produc5716802255202478839_c_nat @ Q5 @ Bs3 ) ) ) ) ) ).

% cstep_def
thf(fact_1116_cstep__def,axiom,
    ( cstep_c_nat
    = ( ^ [Step2: c > nat > c,St3: mappin8329769500858951116_nat_c,Q5: c,Bs3: nat] :
          ( case_o6669308209970876632at_c_c @ ( produc3704122138053201123_nat_c @ ( Step2 @ Q5 @ Bs3 ) @ ( update6790522142438593428_nat_c @ ( product_Pair_c_nat @ Q5 @ Bs3 ) @ ( Step2 @ Q5 @ Bs3 ) @ St3 ) )
          @ ^ [V: c] : ( produc3704122138053201123_nat_c @ V @ St3 )
          @ ( lookup6598465333798894979_nat_c @ St3 @ ( product_Pair_c_nat @ Q5 @ Bs3 ) ) ) ) ) ).

% cstep_def
thf(fact_1117_apfst__conv,axiom,
    ! [F: b > b,X4: b,Y4: option7520157102916957007_c_nat] :
      ( ( produc3151542644641809611_c_nat @ F @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) )
      = ( produc5716802255202478839_c_nat @ ( F @ X4 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_1118_apfst__conv,axiom,
    ! [F: c > c,X4: c,Y4: nat] :
      ( ( produc3709900302687399308_c_nat @ F @ ( product_Pair_c_nat @ X4 @ Y4 ) )
      = ( product_Pair_c_nat @ ( F @ X4 ) @ Y4 ) ) ).

% apfst_conv
thf(fact_1119_apsnd__conv,axiom,
    ! [F: option7520157102916957007_c_nat > option7520157102916957007_c_nat,X4: b,Y4: option7520157102916957007_c_nat] :
      ( ( produc4905112676186601185_nat_b @ F @ ( produc5716802255202478839_c_nat @ X4 @ Y4 ) )
      = ( produc5716802255202478839_c_nat @ X4 @ ( F @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_1120_apsnd__conv,axiom,
    ! [F: nat > nat,X4: c,Y4: nat] :
      ( ( produc8521403278789009504_nat_c @ F @ ( product_Pair_c_nat @ X4 @ Y4 ) )
      = ( product_Pair_c_nat @ X4 @ ( F @ Y4 ) ) ) ).

% apsnd_conv
thf(fact_1121_fst__apfst,axiom,
    ! [F: b > b,X4: produc6689720525226317995_c_nat] :
      ( ( produc6033490199168946105_c_nat @ ( produc6127628685690745977_c_nat @ F @ X4 ) )
      = ( F @ ( produc6033490199168946105_c_nat @ X4 ) ) ) ).

% fst_apfst
thf(fact_1122_fst__apfst,axiom,
    ! [F: b > b,X4: product_prod_b_c] :
      ( ( product_fst_b_c @ ( product_apfst_b_b_c @ F @ X4 ) )
      = ( F @ ( product_fst_b_c @ X4 ) ) ) ).

% fst_apfst
thf(fact_1123_fst__apfst,axiom,
    ! [F: b > b,X4: produc4862256710654508797_c_nat] :
      ( ( produc8816143915943518219_c_nat @ ( produc3151542644641809611_c_nat @ F @ X4 ) )
      = ( F @ ( produc8816143915943518219_c_nat @ X4 ) ) ) ).

% fst_apfst
thf(fact_1124_fst__apsnd,axiom,
    ! [F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat,X4: produc6689720525226317995_c_nat] :
      ( ( produc6033490199168946105_c_nat @ ( produc4918591324248984453_nat_b @ F @ X4 ) )
      = ( produc6033490199168946105_c_nat @ X4 ) ) ).

% fst_apsnd
thf(fact_1125_fst__apsnd,axiom,
    ! [F: c > produc4862256710654508797_c_nat,X4: product_prod_b_c] :
      ( ( produc6033490199168946105_c_nat @ ( produc8789165027643014846_nat_b @ F @ X4 ) )
      = ( product_fst_b_c @ X4 ) ) ).

% fst_apsnd
thf(fact_1126_fst__apsnd,axiom,
    ! [F: option7520157102916957007_c_nat > produc4862256710654508797_c_nat,X4: produc4862256710654508797_c_nat] :
      ( ( produc6033490199168946105_c_nat @ ( produc6752724804963799219_nat_b @ F @ X4 ) )
      = ( produc8816143915943518219_c_nat @ X4 ) ) ).

% fst_apsnd
thf(fact_1127_fst__apsnd,axiom,
    ! [F: produc4862256710654508797_c_nat > c,X4: produc6689720525226317995_c_nat] :
      ( ( product_fst_b_c @ ( produc4536319021487757306at_c_b @ F @ X4 ) )
      = ( produc6033490199168946105_c_nat @ X4 ) ) ).

% fst_apsnd
thf(fact_1128_fst__apsnd,axiom,
    ! [F: c > c,X4: product_prod_b_c] :
      ( ( product_fst_b_c @ ( product_apsnd_c_c_b @ F @ X4 ) )
      = ( product_fst_b_c @ X4 ) ) ).

% fst_apsnd
thf(fact_1129_fst__apsnd,axiom,
    ! [F: option7520157102916957007_c_nat > c,X4: produc4862256710654508797_c_nat] :
      ( ( product_fst_b_c @ ( produc3164437587935750476at_c_b @ F @ X4 ) )
      = ( produc8816143915943518219_c_nat @ X4 ) ) ).

% fst_apsnd
thf(fact_1130_fst__apsnd,axiom,
    ! [F: produc4862256710654508797_c_nat > option7520157102916957007_c_nat,X4: produc6689720525226317995_c_nat] :
      ( ( produc8816143915943518219_c_nat @ ( produc8766251453184775091_nat_b @ F @ X4 ) )
      = ( produc6033490199168946105_c_nat @ X4 ) ) ).

% fst_apsnd
thf(fact_1131_fst__apsnd,axiom,
    ! [F: c > option7520157102916957007_c_nat,X4: product_prod_b_c] :
      ( ( produc8816143915943518219_c_nat @ ( produc6986494041225373676_nat_b @ F @ X4 ) )
      = ( product_fst_b_c @ X4 ) ) ).

% fst_apsnd
thf(fact_1132_fst__apsnd,axiom,
    ! [F: option7520157102916957007_c_nat > option7520157102916957007_c_nat,X4: produc4862256710654508797_c_nat] :
      ( ( produc8816143915943518219_c_nat @ ( produc4905112676186601185_nat_b @ F @ X4 ) )
      = ( produc8816143915943518219_c_nat @ X4 ) ) ).

% fst_apsnd
thf(fact_1133_fst__comp__apsnd,axiom,
    ! [F: produc4862256710654508797_c_nat > produc4862256710654508797_c_nat] :
      ( ( comp_P1437073467185880348_c_nat @ produc6033490199168946105_c_nat @ ( produc4918591324248984453_nat_b @ F ) )
      = produc6033490199168946105_c_nat ) ).

% fst_comp_apsnd
thf(fact_1134_fst__comp__apsnd,axiom,
    ! [F: c > produc4862256710654508797_c_nat] :
      ( ( comp_P6547017101245297847od_b_c @ produc6033490199168946105_c_nat @ ( produc8789165027643014846_nat_b @ F ) )
      = product_fst_b_c ) ).

% fst_comp_apsnd
thf(fact_1135_fst__comp__apsnd,axiom,
    ! [F: option7520157102916957007_c_nat > produc4862256710654508797_c_nat] :
      ( ( comp_P9073800064271228014_c_nat @ produc6033490199168946105_c_nat @ ( produc6752724804963799219_nat_b @ F ) )
      = produc8816143915943518219_c_nat ) ).

% fst_comp_apsnd
thf(fact_1136_fst__comp__apsnd,axiom,
    ! [F: produc4862256710654508797_c_nat > c] :
      ( ( comp_P8916405840549426341_c_nat @ product_fst_b_c @ ( produc4536319021487757306at_c_b @ F ) )
      = produc6033490199168946105_c_nat ) ).

% fst_comp_apsnd
thf(fact_1137_fst__comp__apsnd,axiom,
    ! [F: c > c] :
      ( ( comp_P7908477705713330670od_b_c @ product_fst_b_c @ ( product_apsnd_c_c_b @ F ) )
      = product_fst_b_c ) ).

% fst_comp_apsnd
thf(fact_1138_fst__comp__apsnd,axiom,
    ! [F: option7520157102916957007_c_nat > c] :
      ( ( comp_P6972646978455201015_c_nat @ product_fst_b_c @ ( produc3164437587935750476at_c_b @ F ) )
      = produc8816143915943518219_c_nat ) ).

% fst_comp_apsnd
thf(fact_1139_fst__comp__apsnd,axiom,
    ! [F: produc4862256710654508797_c_nat > option7520157102916957007_c_nat] :
      ( ( comp_P7163099704581481802_c_nat @ produc8816143915943518219_c_nat @ ( produc8766251453184775091_nat_b @ F ) )
      = produc6033490199168946105_c_nat ) ).

% fst_comp_apsnd
thf(fact_1140_fst__comp__apsnd,axiom,
    ! [F: c > option7520157102916957007_c_nat] :
      ( ( comp_P8474455673179243273od_b_c @ produc8816143915943518219_c_nat @ ( produc6986494041225373676_nat_b @ F ) )
      = product_fst_b_c ) ).

% fst_comp_apsnd
thf(fact_1141_fst__comp__apsnd,axiom,
    ! [F: option7520157102916957007_c_nat > option7520157102916957007_c_nat] :
      ( ( comp_P4601747412748468892_c_nat @ produc8816143915943518219_c_nat @ ( produc4905112676186601185_nat_b @ F ) )
      = produc8816143915943518219_c_nat ) ).

% fst_comp_apsnd
thf(fact_1142_fst__comp__apfst,axiom,
    ! [F: b > b] :
      ( ( comp_P1437073467185880348_c_nat @ produc6033490199168946105_c_nat @ ( produc6127628685690745977_c_nat @ F ) )
      = ( comp_b7259836426336601678_c_nat @ F @ produc6033490199168946105_c_nat ) ) ).

% fst_comp_apfst
thf(fact_1143_fst__comp__apfst,axiom,
    ! [F: b > b] :
      ( ( comp_P7908477705713330670od_b_c @ product_fst_b_c @ ( product_apfst_b_b_c @ F ) )
      = ( comp_b1099499051205000453od_b_c @ F @ product_fst_b_c ) ) ).

% fst_comp_apfst
thf(fact_1144_fst__comp__apfst,axiom,
    ! [F: b > b] :
      ( ( comp_P4601747412748468892_c_nat @ produc8816143915943518219_c_nat @ ( produc3151542644641809611_c_nat @ F ) )
      = ( comp_b181056476299970464_c_nat @ F @ produc8816143915943518219_c_nat ) ) ).

% fst_comp_apfst
thf(fact_1145_apfst__apsnd,axiom,
    ! [F: b > b,G: produc4862256710654508797_c_nat > option7520157102916957007_c_nat,X4: produc6689720525226317995_c_nat] :
      ( ( produc3151542644641809611_c_nat @ F @ ( produc8766251453184775091_nat_b @ G @ X4 ) )
      = ( produc5716802255202478839_c_nat @ ( F @ ( produc6033490199168946105_c_nat @ X4 ) ) @ ( G @ ( produc6319817161620345591_c_nat @ X4 ) ) ) ) ).

% apfst_apsnd
thf(fact_1146_apfst__apsnd,axiom,
    ! [F: b > c,G: produc4862256710654508797_c_nat > nat,X4: produc6689720525226317995_c_nat] :
      ( ( produc2365435265877911373_c_nat @ F @ ( produc5685857082087432050_nat_b @ G @ X4 ) )
      = ( product_Pair_c_nat @ ( F @ ( produc6033490199168946105_c_nat @ X4 ) ) @ ( G @ ( produc6319817161620345591_c_nat @ X4 ) ) ) ) ).

% apfst_apsnd
thf(fact_1147_apfst__apsnd,axiom,
    ! [F: b > b,G: c > option7520157102916957007_c_nat,X4: product_prod_b_c] :
      ( ( produc3151542644641809611_c_nat @ F @ ( produc6986494041225373676_nat_b @ G @ X4 ) )
      = ( produc5716802255202478839_c_nat @ ( F @ ( product_fst_b_c @ X4 ) ) @ ( G @ ( product_snd_b_c @ X4 ) ) ) ) ).

% apfst_apsnd
thf(fact_1148_apfst__apsnd,axiom,
    ! [F: b > c,G: c > nat,X4: product_prod_b_c] :
      ( ( produc2365435265877911373_c_nat @ F @ ( produc8334677601181179435_nat_b @ G @ X4 ) )
      = ( product_Pair_c_nat @ ( F @ ( product_fst_b_c @ X4 ) ) @ ( G @ ( product_snd_b_c @ X4 ) ) ) ) ).

% apfst_apsnd
thf(fact_1149_apfst__apsnd,axiom,
    ! [F: b > b,G: option7520157102916957007_c_nat > option7520157102916957007_c_nat,X4: produc4862256710654508797_c_nat] :
      ( ( produc3151542644641809611_c_nat @ F @ ( produc4905112676186601185_nat_b @ G @ X4 ) )
      = ( produc5716802255202478839_c_nat @ ( F @ ( produc8816143915943518219_c_nat @ X4 ) ) @ ( G @ ( produc2435110264752509257_c_nat @ X4 ) ) ) ) ).

% apfst_apsnd
thf(fact_1150_apfst__apsnd,axiom,
    ! [F: b > c,G: option7520157102916957007_c_nat > nat,X4: produc4862256710654508797_c_nat] :
      ( ( produc2365435265877911373_c_nat @ F @ ( produc94040686257552800_nat_b @ G @ X4 ) )
      = ( product_Pair_c_nat @ ( F @ ( produc8816143915943518219_c_nat @ X4 ) ) @ ( G @ ( produc2435110264752509257_c_nat @ X4 ) ) ) ) ).

% apfst_apsnd
thf(fact_1151_apsnd__apfst,axiom,
    ! [F: produc4862256710654508797_c_nat > option7520157102916957007_c_nat,G: b > b,X4: produc6689720525226317995_c_nat] :
      ( ( produc8766251453184775091_nat_b @ F @ ( produc6127628685690745977_c_nat @ G @ X4 ) )
      = ( produc5716802255202478839_c_nat @ ( G @ ( produc6033490199168946105_c_nat @ X4 ) ) @ ( F @ ( produc6319817161620345591_c_nat @ X4 ) ) ) ) ).

% apsnd_apfst
thf(fact_1152_apsnd__apfst,axiom,
    ! [F: produc4862256710654508797_c_nat > nat,G: b > c,X4: produc6689720525226317995_c_nat] :
      ( ( produc5685857082087432051_nat_c @ F @ ( produc501914453448575226_c_nat @ G @ X4 ) )
      = ( product_Pair_c_nat @ ( G @ ( produc6033490199168946105_c_nat @ X4 ) ) @ ( F @ ( produc6319817161620345591_c_nat @ X4 ) ) ) ) ).

% apsnd_apfst
thf(fact_1153_apsnd__apfst,axiom,
    ! [F: c > option7520157102916957007_c_nat,G: b > b,X4: product_prod_b_c] :
      ( ( produc6986494041225373676_nat_b @ F @ ( product_apfst_b_b_c @ G @ X4 ) )
      = ( produc5716802255202478839_c_nat @ ( G @ ( product_fst_b_c @ X4 ) ) @ ( F @ ( product_snd_b_c @ X4 ) ) ) ) ).

% apsnd_apfst
thf(fact_1154_apsnd__apfst,axiom,
    ! [F: c > nat,G: b > c,X4: product_prod_b_c] :
      ( ( produc8334677601181179436_nat_c @ F @ ( product_apfst_b_c_c @ G @ X4 ) )
      = ( product_Pair_c_nat @ ( G @ ( product_fst_b_c @ X4 ) ) @ ( F @ ( product_snd_b_c @ X4 ) ) ) ) ).

% apsnd_apfst
thf(fact_1155_apsnd__apfst,axiom,
    ! [F: option7520157102916957007_c_nat > option7520157102916957007_c_nat,G: b > b,X4: produc4862256710654508797_c_nat] :
      ( ( produc4905112676186601185_nat_b @ F @ ( produc3151542644641809611_c_nat @ G @ X4 ) )
      = ( produc5716802255202478839_c_nat @ ( G @ ( produc8816143915943518219_c_nat @ X4 ) ) @ ( F @ ( produc2435110264752509257_c_nat @ X4 ) ) ) ) ).

% apsnd_apfst
thf(fact_1156_apsnd__apfst,axiom,
    ! [F: option7520157102916957007_c_nat > nat,G: b > c,X4: produc4862256710654508797_c_nat] :
      ( ( produc94040686257552801_nat_c @ F @ ( produc6524259015346455116_c_nat @ G @ X4 ) )
      = ( product_Pair_c_nat @ ( G @ ( produc8816143915943518219_c_nat @ X4 ) ) @ ( F @ ( produc2435110264752509257_c_nat @ X4 ) ) ) ) ).

% apsnd_apfst
thf(fact_1157_apfst__convE,axiom,
    ! [Q3: produc4862256710654508797_c_nat,F: b > b,P: produc4862256710654508797_c_nat] :
      ( ( Q3
        = ( produc3151542644641809611_c_nat @ F @ P ) )
     => ~ ! [X: b,Y: option7520157102916957007_c_nat] :
            ( ( P
              = ( produc5716802255202478839_c_nat @ X @ Y ) )
           => ( Q3
             != ( produc5716802255202478839_c_nat @ ( F @ X ) @ Y ) ) ) ) ).

% apfst_convE
thf(fact_1158_apfst__convE,axiom,
    ! [Q3: product_prod_c_nat,F: c > c,P: product_prod_c_nat] :
      ( ( Q3
        = ( produc3709900302687399308_c_nat @ F @ P ) )
     => ~ ! [X: c,Y: nat] :
            ( ( P
              = ( product_Pair_c_nat @ X @ Y ) )
           => ( Q3
             != ( product_Pair_c_nat @ ( F @ X ) @ Y ) ) ) ) ).

% apfst_convE
thf(fact_1159_map__rec,axiom,
    ( map_Pr6200325787298368846_nat_b
    = ( ^ [F3: produc6689720525226317995_c_nat > b] :
          ( rec_li3848129933990450042_c_nat @ nil_b
          @ ^ [X3: produc6689720525226317995_c_nat,Uu: list_P7417839048565863355_c_nat] : ( cons_b @ ( F3 @ X3 ) ) ) ) ) ).

% map_rec
thf(fact_1160_map__rec,axiom,
    ( map_Pr8222292247188198875_b_c_b
    = ( ^ [F3: product_prod_b_c > b] :
          ( rec_li972666565249325273od_b_c @ nil_b
          @ ^ [X3: product_prod_b_c,Uu: list_P903359562653991662od_b_c] : ( cons_b @ ( F3 @ X3 ) ) ) ) ) ).

% map_rec
thf(fact_1161_minus__coset__filter,axiom,
    ! [A4: set_b,Xs: list_b] :
      ( ( minus_minus_set_b @ A4 @ ( coset_b @ Xs ) )
      = ( set_b2
        @ ( filter_b
          @ ^ [X3: b] : ( member_b @ X3 @ A4 )
          @ Xs ) ) ) ).

% minus_coset_filter
thf(fact_1162_union__coset__filter,axiom,
    ! [Xs: list_b,A4: set_b] :
      ( ( sup_sup_set_b @ ( coset_b @ Xs ) @ A4 )
      = ( coset_b
        @ ( filter_b
          @ ^ [X3: b] :
              ~ ( member_b @ X3 @ A4 )
          @ Xs ) ) ) ).

% union_coset_filter
thf(fact_1163_map__of__delete__aux_H,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,K2: b] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs ) )
     => ( ( map_of556380330368225625_c_nat @ ( delete9177780190517604182_c_nat @ K2 @ Xs ) )
        = ( fun_up6679455600583992207_c_nat @ ( map_of556380330368225625_c_nat @ Xs ) @ K2 @ none_P8487441334512977628_c_nat ) ) ) ).

% map_of_delete_aux'
thf(fact_1164_map__of__delete__aux_H,axiom,
    ! [Xs: list_P903359562653991662od_b_c,K2: b] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs ) )
     => ( ( map_of_b_c @ ( delete_aux_b_c @ K2 @ Xs ) )
        = ( fun_upd_b_option_c @ ( map_of_b_c @ Xs ) @ K2 @ none_c ) ) ) ).

% map_of_delete_aux'
thf(fact_1165_map__of__delete__aux_H,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,K2: b] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs ) )
     => ( ( map_of2598500733885366699_c_nat @ ( delete7176321171147170984_c_nat @ K2 @ Xs ) )
        = ( fun_up2348109198587530721_c_nat @ ( map_of2598500733885366699_c_nat @ Xs ) @ K2 @ none_o1329286346417566254_c_nat ) ) ) ).

% map_of_delete_aux'
thf(fact_1166_map__fun__upd,axiom,
    ! [Y4: produc6689720525226317995_c_nat,Xs: list_P7417839048565863355_c_nat,F: produc6689720525226317995_c_nat > b,V4: b] :
      ( ~ ( member1522478204950355394_c_nat @ Y4 @ ( set_Pr5807476769692288598_c_nat @ Xs ) )
     => ( ( map_Pr6200325787298368846_nat_b @ ( fun_up3995235727163668821_nat_b @ F @ Y4 @ V4 ) @ Xs )
        = ( map_Pr6200325787298368846_nat_b @ F @ Xs ) ) ) ).

% map_fun_upd
thf(fact_1167_map__fun__upd,axiom,
    ! [Y4: product_prod_b_c,Xs: list_P903359562653991662od_b_c,F: product_prod_b_c > b,V4: b] :
      ( ~ ( member7862447941013992593od_b_c @ Y4 @ ( set_Product_prod_b_c2 @ Xs ) )
     => ( ( map_Pr8222292247188198875_b_c_b @ ( fun_up263639303302703380_b_c_b @ F @ Y4 @ V4 ) @ Xs )
        = ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) ) ) ).

% map_fun_upd
thf(fact_1168_fun__upd__None__restrict,axiom,
    ! [X4: b,D2: set_b,M: b > option2970150418924381261_c_nat] :
      ( ( ( member_b @ X4 @ D2 )
       => ( ( fun_up6679455600583992207_c_nat @ ( restri4430789530154927564_c_nat @ M @ D2 ) @ X4 @ none_P8487441334512977628_c_nat )
          = ( restri4430789530154927564_c_nat @ M @ ( minus_minus_set_b @ D2 @ ( insert_b @ X4 @ bot_bot_set_b ) ) ) ) )
      & ( ~ ( member_b @ X4 @ D2 )
       => ( ( fun_up6679455600583992207_c_nat @ ( restri4430789530154927564_c_nat @ M @ D2 ) @ X4 @ none_P8487441334512977628_c_nat )
          = ( restri4430789530154927564_c_nat @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_1169_map__of_Osimps_I2_J,axiom,
    ! [P: produc6689720525226317995_c_nat,Ps: list_P7417839048565863355_c_nat] :
      ( ( map_of556380330368225625_c_nat @ ( cons_P7294201633519847029_c_nat @ P @ Ps ) )
      = ( fun_up6679455600583992207_c_nat @ ( map_of556380330368225625_c_nat @ Ps ) @ ( produc6033490199168946105_c_nat @ P ) @ ( some_P2720002978653898840_c_nat @ ( produc6319817161620345591_c_nat @ P ) ) ) ) ).

% map_of.simps(2)
thf(fact_1170_map__of_Osimps_I2_J,axiom,
    ! [P: product_prod_b_c,Ps: list_P903359562653991662od_b_c] :
      ( ( map_of_b_c @ ( cons_P4529483553340347422od_b_c @ P @ Ps ) )
      = ( fun_upd_b_option_c @ ( map_of_b_c @ Ps ) @ ( product_fst_b_c @ P ) @ ( some_c @ ( product_snd_b_c @ P ) ) ) ) ).

% map_of.simps(2)
thf(fact_1171_map__of_Osimps_I2_J,axiom,
    ! [P: produc4862256710654508797_c_nat,Ps: list_P8155366108056616461_c_nat] :
      ( ( map_of2598500733885366699_c_nat @ ( cons_P7928565014262433479_c_nat @ P @ Ps ) )
      = ( fun_up2348109198587530721_c_nat @ ( map_of2598500733885366699_c_nat @ Ps ) @ ( produc8816143915943518219_c_nat @ P ) @ ( some_o7552056839342673834_c_nat @ ( produc2435110264752509257_c_nat @ P ) ) ) ) ).

% map_of.simps(2)
thf(fact_1172_map__of__delete__aux,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,K2: b,K5: b] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs ) )
     => ( ( map_of556380330368225625_c_nat @ ( delete9177780190517604182_c_nat @ K2 @ Xs ) @ K5 )
        = ( fun_up6679455600583992207_c_nat @ ( map_of556380330368225625_c_nat @ Xs ) @ K2 @ none_P8487441334512977628_c_nat @ K5 ) ) ) ).

% map_of_delete_aux
thf(fact_1173_map__of__delete__aux,axiom,
    ! [Xs: list_P903359562653991662od_b_c,K2: b,K5: b] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs ) )
     => ( ( map_of_b_c @ ( delete_aux_b_c @ K2 @ Xs ) @ K5 )
        = ( fun_upd_b_option_c @ ( map_of_b_c @ Xs ) @ K2 @ none_c @ K5 ) ) ) ).

% map_of_delete_aux
thf(fact_1174_map__of__delete__aux,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,K2: b,K5: b] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs ) )
     => ( ( map_of2598500733885366699_c_nat @ ( delete7176321171147170984_c_nat @ K2 @ Xs ) @ K5 )
        = ( fun_up2348109198587530721_c_nat @ ( map_of2598500733885366699_c_nat @ Xs ) @ K2 @ none_o1329286346417566254_c_nat @ K5 ) ) ) ).

% map_of_delete_aux
thf(fact_1175_graph__map__upd,axiom,
    ! [M: b > option6424319895471668383_c_nat,K2: b,V4: option7520157102916957007_c_nat] :
      ( ( graph_5280084741242382672_c_nat @ ( fun_up2348109198587530721_c_nat @ M @ K2 @ ( some_o7552056839342673834_c_nat @ V4 ) ) )
      = ( insert8034245793825879341_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ V4 ) @ ( graph_5280084741242382672_c_nat @ ( fun_up2348109198587530721_c_nat @ M @ K2 @ none_o1329286346417566254_c_nat ) ) ) ) ).

% graph_map_upd
thf(fact_1176_graph__map__upd,axiom,
    ! [M: c > option_nat,K2: c,V4: nat] :
      ( ( graph_c_nat @ ( fun_upd_c_option_nat @ M @ K2 @ ( some_nat @ V4 ) ) )
      = ( insert1302120044651762863_c_nat @ ( product_Pair_c_nat @ K2 @ V4 ) @ ( graph_c_nat @ ( fun_upd_c_option_nat @ M @ K2 @ none_nat ) ) ) ) ).

% graph_map_upd
thf(fact_1177_fun__upd__None__if__notin__dom,axiom,
    ! [K2: b,M: b > option2970150418924381261_c_nat] :
      ( ~ ( member_b @ K2 @ ( dom_b_2330607713053947690_c_nat @ M ) )
     => ( ( fun_up6679455600583992207_c_nat @ M @ K2 @ none_P8487441334512977628_c_nat )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_1178_in__graphD,axiom,
    ! [K2: b,V4: option7520157102916957007_c_nat,M: b > option6424319895471668383_c_nat] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ V4 ) @ ( graph_5280084741242382672_c_nat @ M ) )
     => ( ( M @ K2 )
        = ( some_o7552056839342673834_c_nat @ V4 ) ) ) ).

% in_graphD
thf(fact_1179_in__graphD,axiom,
    ! [K2: c,V4: nat,M: c > option_nat] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ K2 @ V4 ) @ ( graph_c_nat @ M ) )
     => ( ( M @ K2 )
        = ( some_nat @ V4 ) ) ) ).

% in_graphD
thf(fact_1180_in__graphI,axiom,
    ! [M: b > option6424319895471668383_c_nat,K2: b,V4: option7520157102916957007_c_nat] :
      ( ( ( M @ K2 )
        = ( some_o7552056839342673834_c_nat @ V4 ) )
     => ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ V4 ) @ ( graph_5280084741242382672_c_nat @ M ) ) ) ).

% in_graphI
thf(fact_1181_in__graphI,axiom,
    ! [M: c > option_nat,K2: c,V4: nat] :
      ( ( ( M @ K2 )
        = ( some_nat @ V4 ) )
     => ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ K2 @ V4 ) @ ( graph_c_nat @ M ) ) ) ).

% in_graphI
thf(fact_1182_graph__eq__to__snd__dom,axiom,
    ( graph_5280084741242382672_c_nat
    = ( ^ [M2: b > option6424319895471668383_c_nat] :
          ( image_8746285289878263735_c_nat
          @ ^ [X3: b] : ( produc5716802255202478839_c_nat @ X3 @ ( the_op8892588699428011963_c_nat @ ( M2 @ X3 ) ) )
          @ ( dom_b_7471247815972823164_c_nat @ M2 ) ) ) ) ).

% graph_eq_to_snd_dom
thf(fact_1183_graph__eq__to__snd__dom,axiom,
    ( graph_c_nat
    = ( ^ [M2: c > option_nat] :
          ( image_3285331874319137594_c_nat
          @ ^ [X3: c] : ( product_Pair_c_nat @ X3 @ ( the_nat @ ( M2 @ X3 ) ) )
          @ ( dom_c_nat @ M2 ) ) ) ) ).

% graph_eq_to_snd_dom
thf(fact_1184_fst__graph__eq__dom,axiom,
    ! [M: b > option2970150418924381261_c_nat] :
      ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( graph_4208730648046144510_c_nat @ M ) )
      = ( dom_b_2330607713053947690_c_nat @ M ) ) ).

% fst_graph_eq_dom
thf(fact_1185_fst__graph__eq__dom,axiom,
    ! [M: b > option_c] :
      ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( graph_b_c @ M ) )
      = ( dom_b_c @ M ) ) ).

% fst_graph_eq_dom
thf(fact_1186_fst__graph__eq__dom,axiom,
    ! [M: b > option6424319895471668383_c_nat] :
      ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( graph_5280084741242382672_c_nat @ M ) )
      = ( dom_b_7471247815972823164_c_nat @ M ) ) ).

% fst_graph_eq_dom
thf(fact_1187_domIff,axiom,
    ! [A: b,M: b > option2970150418924381261_c_nat] :
      ( ( member_b @ A @ ( dom_b_2330607713053947690_c_nat @ M ) )
      = ( ( M @ A )
       != none_P8487441334512977628_c_nat ) ) ).

% domIff
thf(fact_1188_Compl__eq,axiom,
    ( uminus_uminus_set_b
    = ( ^ [A6: set_b] :
          ( collect_b
          @ ^ [X3: b] :
              ~ ( member_b @ X3 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_1189_graph__restrictD_I1_J,axiom,
    ! [K2: b,V4: option7520157102916957007_c_nat,M: b > option6424319895471668383_c_nat,A4: set_b] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ V4 ) @ ( graph_5280084741242382672_c_nat @ ( restri8862894144314794270_c_nat @ M @ A4 ) ) )
     => ( member_b @ K2 @ A4 ) ) ).

% graph_restrictD(1)
thf(fact_1190_graph__restrictD_I1_J,axiom,
    ! [K2: c,V4: nat,M: c > option_nat,A4: set_c] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ K2 @ V4 ) @ ( graph_c_nat @ ( restrict_map_c_nat @ M @ A4 ) ) )
     => ( member_c @ K2 @ A4 ) ) ).

% graph_restrictD(1)
thf(fact_1191_graph__restrictD_I2_J,axiom,
    ! [K2: b,V4: option7520157102916957007_c_nat,M: b > option6424319895471668383_c_nat,A4: set_b] :
      ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ K2 @ V4 ) @ ( graph_5280084741242382672_c_nat @ ( restri8862894144314794270_c_nat @ M @ A4 ) ) )
     => ( ( M @ K2 )
        = ( some_o7552056839342673834_c_nat @ V4 ) ) ) ).

% graph_restrictD(2)
thf(fact_1192_graph__restrictD_I2_J,axiom,
    ! [K2: c,V4: nat,M: c > option_nat,A4: set_c] :
      ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ K2 @ V4 ) @ ( graph_c_nat @ ( restrict_map_c_nat @ M @ A4 ) ) )
     => ( ( M @ K2 )
        = ( some_nat @ V4 ) ) ) ).

% graph_restrictD(2)
thf(fact_1193_graph__fun__upd__None,axiom,
    ! [M: b > option2970150418924381261_c_nat,K2: b] :
      ( ( graph_4208730648046144510_c_nat @ ( fun_up6679455600583992207_c_nat @ M @ K2 @ none_P8487441334512977628_c_nat ) )
      = ( collec8595031719707015680_c_nat
        @ ^ [E: produc6689720525226317995_c_nat] :
            ( ( member1522478204950355394_c_nat @ E @ ( graph_4208730648046144510_c_nat @ M ) )
            & ( ( produc6033490199168946105_c_nat @ E )
             != K2 ) ) ) ) ).

% graph_fun_upd_None
thf(fact_1194_graph__fun__upd__None,axiom,
    ! [M: b > option_c,K2: b] :
      ( ( graph_b_c @ ( fun_upd_b_option_c @ M @ K2 @ none_c ) )
      = ( collec548942224018234067od_b_c
        @ ^ [E: product_prod_b_c] :
            ( ( member7862447941013992593od_b_c @ E @ ( graph_b_c @ M ) )
            & ( ( product_fst_b_c @ E )
             != K2 ) ) ) ) ).

% graph_fun_upd_None
thf(fact_1195_graph__fun__upd__None,axiom,
    ! [M: b > option6424319895471668383_c_nat,K2: b] :
      ( ( graph_5280084741242382672_c_nat @ ( fun_up2348109198587530721_c_nat @ M @ K2 @ none_o1329286346417566254_c_nat ) )
      = ( collec4170641529751027026_c_nat
        @ ^ [E: produc4862256710654508797_c_nat] :
            ( ( member7562873241046315796_c_nat @ E @ ( graph_5280084741242382672_c_nat @ M ) )
            & ( ( produc8816143915943518219_c_nat @ E )
             != K2 ) ) ) ) ).

% graph_fun_upd_None
thf(fact_1196_dom__map__of__conv__image__fst,axiom,
    ! [Xys: list_P7417839048565863355_c_nat] :
      ( ( dom_b_2330607713053947690_c_nat @ ( map_of556380330368225625_c_nat @ Xys ) )
      = ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Xys ) ) ) ).

% dom_map_of_conv_image_fst
thf(fact_1197_dom__map__of__conv__image__fst,axiom,
    ! [Xys: list_P903359562653991662od_b_c] :
      ( ( dom_b_c @ ( map_of_b_c @ Xys ) )
      = ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Xys ) ) ) ).

% dom_map_of_conv_image_fst
thf(fact_1198_dom__map__of__conv__image__fst,axiom,
    ! [Xys: list_P8155366108056616461_c_nat] :
      ( ( dom_b_7471247815972823164_c_nat @ ( map_of2598500733885366699_c_nat @ Xys ) )
      = ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Xys ) ) ) ).

% dom_map_of_conv_image_fst
thf(fact_1199_graph__map__of__if__distinct__dom,axiom,
    ! [Al: list_P7417839048565863355_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Al ) )
     => ( ( graph_4208730648046144510_c_nat @ ( map_of556380330368225625_c_nat @ Al ) )
        = ( set_Pr5807476769692288598_c_nat @ Al ) ) ) ).

% graph_map_of_if_distinct_dom
thf(fact_1200_graph__map__of__if__distinct__dom,axiom,
    ! [Al: list_P903359562653991662od_b_c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Al ) )
     => ( ( graph_b_c @ ( map_of_b_c @ Al ) )
        = ( set_Product_prod_b_c2 @ Al ) ) ) ).

% graph_map_of_if_distinct_dom
thf(fact_1201_graph__map__of__if__distinct__dom,axiom,
    ! [Al: list_P8155366108056616461_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Al ) )
     => ( ( graph_5280084741242382672_c_nat @ ( map_of2598500733885366699_c_nat @ Al ) )
        = ( set_Pr5735414202003984296_c_nat @ Al ) ) ) ).

% graph_map_of_if_distinct_dom
thf(fact_1202_map__of__map__keys,axiom,
    ! [Xs: list_b,M: b > option6424319895471668383_c_nat] :
      ( ( ( set_b2 @ Xs )
        = ( dom_b_7471247815972823164_c_nat @ M ) )
     => ( ( map_of2598500733885366699_c_nat
          @ ( map_b_6442276761558007352_c_nat
            @ ^ [K: b] : ( produc5716802255202478839_c_nat @ K @ ( the_op8892588699428011963_c_nat @ ( M @ K ) ) )
            @ Xs ) )
        = M ) ) ).

% map_of_map_keys
thf(fact_1203_map__of__map__keys,axiom,
    ! [Xs: list_c,M: c > option_nat] :
      ( ( ( set_c2 @ Xs )
        = ( dom_c_nat @ M ) )
     => ( ( map_of_c_nat
          @ ( map_c_3232936969464635451_c_nat
            @ ^ [K: c] : ( product_Pair_c_nat @ K @ ( the_nat @ ( M @ K ) ) )
            @ Xs ) )
        = M ) ) ).

% map_of_map_keys
thf(fact_1204_dom__merge,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,Ys: list_P7417839048565863355_c_nat] :
      ( ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ ( merge_18683747638978007_c_nat @ Xs @ Ys ) ) )
      = ( sup_sup_set_b @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Xs ) ) @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Ys ) ) ) ) ).

% dom_merge
thf(fact_1205_dom__merge,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( merge_b_c @ Xs @ Ys ) ) )
      = ( sup_sup_set_b @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Xs ) ) @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Ys ) ) ) ) ).

% dom_merge
thf(fact_1206_dom__merge,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,Ys: list_P8155366108056616461_c_nat] :
      ( ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ ( merge_2858581043212764201_c_nat @ Xs @ Ys ) ) )
      = ( sup_sup_set_b @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Xs ) ) @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Ys ) ) ) ) ).

% dom_merge
thf(fact_1207_uminus__set__def,axiom,
    ( uminus_uminus_set_b
    = ( ^ [A6: set_b] :
          ( collect_b
          @ ( uminus_uminus_b_o
            @ ^ [X3: b] : ( member_b @ X3 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_1208_distinct__merge,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,Ys: list_P7417839048565863355_c_nat] :
      ( ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ Xs ) )
     => ( distinct_b @ ( map_Pr6200325787298368846_nat_b @ produc6033490199168946105_c_nat @ ( merge_18683747638978007_c_nat @ Xs @ Ys ) ) ) ) ).

% distinct_merge
thf(fact_1209_distinct__merge,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P903359562653991662od_b_c] :
      ( ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ Xs ) )
     => ( distinct_b @ ( map_Pr8222292247188198875_b_c_b @ product_fst_b_c @ ( merge_b_c @ Xs @ Ys ) ) ) ) ).

% distinct_merge
thf(fact_1210_distinct__merge,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,Ys: list_P8155366108056616461_c_nat] :
      ( ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ Xs ) )
     => ( distinct_b @ ( map_Pr418519014202471036_nat_b @ produc8816143915943518219_c_nat @ ( merge_2858581043212764201_c_nat @ Xs @ Ys ) ) ) ) ).

% distinct_merge
thf(fact_1211_image__subsetI,axiom,
    ! [A4: set_b,F: b > b,B5: set_b] :
      ( ! [X: b] :
          ( ( member_b @ X @ A4 )
         => ( member_b @ ( F @ X ) @ B5 ) )
     => ( ord_less_eq_set_b @ ( image_b_b @ F @ A4 ) @ B5 ) ) ).

% image_subsetI
thf(fact_1212_inj__on__image__mem__iff,axiom,
    ! [F: b > b,B5: set_b,A: b,A4: set_b] :
      ( ( inj_on_b_b @ F @ B5 )
     => ( ( member_b @ A @ B5 )
       => ( ( ord_less_eq_set_b @ A4 @ B5 )
         => ( ( member_b @ ( F @ A ) @ ( image_b_b @ F @ A4 ) )
            = ( member_b @ A @ A4 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1213_subrelI,axiom,
    ! [R2: set_Pr8806432033423503795_c_nat,S2: set_Pr8806432033423503795_c_nat] :
      ( ! [X: b,Y: option7520157102916957007_c_nat] :
          ( ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X @ Y ) @ R2 )
         => ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X @ Y ) @ S2 ) )
     => ( ord_le7634486248364433939_c_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_1214_subrelI,axiom,
    ! [R2: set_Pr6903500605879609269_c_nat,S2: set_Pr6903500605879609269_c_nat] :
      ( ! [X: c,Y: nat] :
          ( ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X @ Y ) @ R2 )
         => ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X @ Y ) @ S2 ) )
     => ( ord_le1411700432677832725_c_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_1215_subset__Collect__iff,axiom,
    ! [B5: set_b,A4: set_b,P2: b > $o] :
      ( ( ord_less_eq_set_b @ B5 @ A4 )
     => ( ( ord_less_eq_set_b @ B5
          @ ( collect_b
            @ ^ [X3: b] :
                ( ( member_b @ X3 @ A4 )
                & ( P2 @ X3 ) ) ) )
        = ( ! [X3: b] :
              ( ( member_b @ X3 @ B5 )
             => ( P2 @ X3 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1216_subset__CollectI,axiom,
    ! [B5: set_b,A4: set_b,Q2: b > $o,P2: b > $o] :
      ( ( ord_less_eq_set_b @ B5 @ A4 )
     => ( ! [X: b] :
            ( ( member_b @ X @ B5 )
           => ( ( Q2 @ X )
             => ( P2 @ X ) ) )
       => ( ord_less_eq_set_b
          @ ( collect_b
            @ ^ [X3: b] :
                ( ( member_b @ X3 @ B5 )
                & ( Q2 @ X3 ) ) )
          @ ( collect_b
            @ ^ [X3: b] :
                ( ( member_b @ X3 @ A4 )
                & ( P2 @ X3 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1217_prop__restrict,axiom,
    ! [X4: b,Z5: set_b,X7: set_b,P2: b > $o] :
      ( ( member_b @ X4 @ Z5 )
     => ( ( ord_less_eq_set_b @ Z5
          @ ( collect_b
            @ ^ [X3: b] :
                ( ( member_b @ X3 @ X7 )
                & ( P2 @ X3 ) ) ) )
       => ( P2 @ X4 ) ) ) ).

% prop_restrict
thf(fact_1218_Collect__restrict,axiom,
    ! [X7: set_b,P2: b > $o] :
      ( ord_less_eq_set_b
      @ ( collect_b
        @ ^ [X3: b] :
            ( ( member_b @ X3 @ X7 )
            & ( P2 @ X3 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_1219_Collect__subset,axiom,
    ! [A4: set_b,P2: b > $o] :
      ( ord_less_eq_set_b
      @ ( collect_b
        @ ^ [X3: b] :
            ( ( member_b @ X3 @ A4 )
            & ( P2 @ X3 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_1220_insert__subsetI,axiom,
    ! [X4: b,A4: set_b,X7: set_b] :
      ( ( member_b @ X4 @ A4 )
     => ( ( ord_less_eq_set_b @ X7 @ A4 )
       => ( ord_less_eq_set_b @ ( insert_b @ X4 @ X7 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_1221_subset__emptyI,axiom,
    ! [A4: set_b] :
      ( ! [X: b] :
          ~ ( member_b @ X @ A4 )
     => ( ord_less_eq_set_b @ A4 @ bot_bot_set_b ) ) ).

% subset_emptyI
thf(fact_1222_Collect__split__mono__strong,axiom,
    ! [X7: set_b,A4: set_Pr93524137074774369_c_nat,Y8: set_Pr8806432033423503795_c_nat,P2: b > produc4862256710654508797_c_nat > $o,Q2: b > produc4862256710654508797_c_nat > $o] :
      ( ( X7
        = ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ A4 ) )
     => ( ( Y8
          = ( image_665133137765360559_c_nat @ produc6319817161620345591_c_nat @ A4 ) )
       => ( ! [X: b] :
              ( ( member_b @ X @ X7 )
             => ! [Xa2: produc4862256710654508797_c_nat] :
                  ( ( member7562873241046315796_c_nat @ Xa2 @ Y8 )
                 => ( ( P2 @ X @ Xa2 )
                   => ( Q2 @ X @ Xa2 ) ) ) )
         => ( ( ord_le4137630889022497217_c_nat @ A4 @ ( collec8595031719707015680_c_nat @ ( produc6484547981188837958_nat_o @ P2 ) ) )
           => ( ord_le4137630889022497217_c_nat @ A4 @ ( collec8595031719707015680_c_nat @ ( produc6484547981188837958_nat_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_1223_Collect__split__mono__strong,axiom,
    ! [X7: set_b,A4: set_Product_prod_b_c,Y8: set_c,P2: b > c > $o,Q2: b > c > $o] :
      ( ( X7
        = ( image_7762865866877615260_b_c_b @ product_fst_b_c @ A4 ) )
     => ( ( Y8
          = ( image_7762865866877615261_b_c_c @ product_snd_b_c @ A4 ) )
       => ( ! [X: b] :
              ( ( member_b @ X @ X7 )
             => ! [Xa2: c] :
                  ( ( member_c @ Xa2 @ Y8 )
                 => ( ( P2 @ X @ Xa2 )
                   => ( Q2 @ X @ Xa2 ) ) ) )
         => ( ( ord_le253122037897321832od_b_c @ A4 @ ( collec548942224018234067od_b_c @ ( produc1873627588981547243_b_c_o @ P2 ) ) )
           => ( ord_le253122037897321832od_b_c @ A4 @ ( collec548942224018234067od_b_c @ ( produc1873627588981547243_b_c_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_1224_Collect__split__mono__strong,axiom,
    ! [X7: set_b,A4: set_Pr8806432033423503795_c_nat,Y8: set_op6398572775360143365_c_nat,P2: b > option7520157102916957007_c_nat > $o,Q2: b > option7520157102916957007_c_nat > $o] :
      ( ( X7
        = ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ A4 ) )
     => ( ( Y8
          = ( image_5441275730643634515_c_nat @ produc2435110264752509257_c_nat @ A4 ) )
       => ( ! [X: b] :
              ( ( member_b @ X @ X7 )
             => ! [Xa2: option7520157102916957007_c_nat] :
                  ( ( member3208607818295432806_c_nat @ Xa2 @ Y8 )
                 => ( ( P2 @ X @ Xa2 )
                   => ( Q2 @ X @ Xa2 ) ) ) )
         => ( ( ord_le7634486248364433939_c_nat @ A4 @ ( collec4170641529751027026_c_nat @ ( produc2544960137906461044_nat_o @ P2 ) ) )
           => ( ord_le7634486248364433939_c_nat @ A4 @ ( collec4170641529751027026_c_nat @ ( produc2544960137906461044_nat_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_1225_dom__delete__subset,axiom,
    ! [K2: b,Al: list_P7417839048565863355_c_nat] : ( ord_less_eq_set_b @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ ( delete1755679665865935934_c_nat @ K2 @ Al ) ) ) @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Al ) ) ) ).

% dom_delete_subset
thf(fact_1226_dom__delete__subset,axiom,
    ! [K2: b,Al: list_P903359562653991662od_b_c] : ( ord_less_eq_set_b @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( delete_b_c @ K2 @ Al ) ) ) @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Al ) ) ) ).

% dom_delete_subset
thf(fact_1227_dom__delete__subset,axiom,
    ! [K2: b,Al: list_P8155366108056616461_c_nat] : ( ord_less_eq_set_b @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ ( delete4993458036327213456_c_nat @ K2 @ Al ) ) ) @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Al ) ) ) ).

% dom_delete_subset
thf(fact_1228_dom__compose,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,Ys: list_P5405424656961669729_c_nat] : ( ord_less_eq_set_b @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ ( compos1751932118770894235_c_nat @ Xs @ Ys ) ) ) @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Xs ) ) ) ).

% dom_compose
thf(fact_1229_dom__compose,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P4207096184057010108_c_nat] : ( ord_less_eq_set_b @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ ( compos5151074131599017758_c_nat @ Xs @ Ys ) ) ) @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Xs ) ) ) ).

% dom_compose
thf(fact_1230_dom__compose,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,Ys: list_P2361265211252882099_c_nat] : ( ord_less_eq_set_b @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ ( compos6055679653055957741_c_nat @ Xs @ Ys ) ) ) @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Xs ) ) ) ).

% dom_compose
thf(fact_1231_dom__compose,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,Ys: list_P7599720138408045768_nat_c] : ( ord_less_eq_set_b @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( compos4753030616485652194_nat_c @ Xs @ Ys ) ) ) @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Xs ) ) ) ).

% dom_compose
thf(fact_1232_dom__compose,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P267710562049004973od_c_c] : ( ord_less_eq_set_b @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( compose_b_c_c @ Xs @ Ys ) ) ) @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Xs ) ) ) ).

% dom_compose
thf(fact_1233_dom__compose,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,Ys: list_P486377699747300854_nat_c] : ( ord_less_eq_set_b @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ ( compos2116506346773900048_nat_c @ Xs @ Ys ) ) ) @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Xs ) ) ) ).

% dom_compose
thf(fact_1234_dom__compose,axiom,
    ! [Xs: list_P7417839048565863355_c_nat,Ys: list_P6797689553898924723_c_nat] : ( ord_less_eq_set_b @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ ( compos3540800493588629485_c_nat @ Xs @ Ys ) ) ) @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ ( set_Pr5807476769692288598_c_nat @ Xs ) ) ) ).

% dom_compose
thf(fact_1235_dom__compose,axiom,
    ! [Xs: list_P903359562653991662od_b_c,Ys: list_P5760860764591065870_c_nat] : ( ord_less_eq_set_b @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ ( compos1359844409926042736_c_nat @ Xs @ Ys ) ) ) @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ ( set_Product_prod_b_c2 @ Xs ) ) ) ).

% dom_compose
thf(fact_1236_dom__compose,axiom,
    ! [Xs: list_P8155366108056616461_c_nat,Ys: list_P2602636364039601413_c_nat] : ( ord_less_eq_set_b @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ ( compos1451257573582057279_c_nat @ Xs @ Ys ) ) ) @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ ( set_Pr5735414202003984296_c_nat @ Xs ) ) ) ).

% dom_compose
thf(fact_1237_subset__fst__snd,axiom,
    ! [A4: set_Pr93524137074774369_c_nat] :
      ( ord_le4137630889022497217_c_nat @ A4
      @ ( produc237220987571061300_c_nat @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ A4 )
        @ ^ [Uu: b] : ( image_665133137765360559_c_nat @ produc6319817161620345591_c_nat @ A4 ) ) ) ).

% subset_fst_snd
thf(fact_1238_subset__fst__snd,axiom,
    ! [A4: set_Product_prod_b_c] :
      ( ord_le253122037897321832od_b_c @ A4
      @ ( product_Sigma_b_c @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ A4 )
        @ ^ [Uu: b] : ( image_7762865866877615261_b_c_c @ product_snd_b_c @ A4 ) ) ) ).

% subset_fst_snd
thf(fact_1239_subset__fst__snd,axiom,
    ! [A4: set_Pr8806432033423503795_c_nat] :
      ( ord_le7634486248364433939_c_nat @ A4
      @ ( produc1892124643397453190_c_nat @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ A4 )
        @ ^ [Uu: b] : ( image_5441275730643634515_c_nat @ produc2435110264752509257_c_nat @ A4 ) ) ) ).

% subset_fst_snd
thf(fact_1240_subset__fst__imageI,axiom,
    ! [A4: set_b,B5: set_Pr8806432033423503795_c_nat,S: set_Pr93524137074774369_c_nat,Y4: produc4862256710654508797_c_nat] :
      ( ( ord_le4137630889022497217_c_nat
        @ ( produc237220987571061300_c_nat @ A4
          @ ^ [Uu: b] : B5 )
        @ S )
     => ( ( member7562873241046315796_c_nat @ Y4 @ B5 )
       => ( ord_less_eq_set_b @ A4 @ ( image_9118790927800078285_nat_b @ produc6033490199168946105_c_nat @ S ) ) ) ) ).

% subset_fst_imageI
thf(fact_1241_subset__fst__imageI,axiom,
    ! [A4: set_b,B5: set_c,S: set_Product_prod_b_c,Y4: c] :
      ( ( ord_le253122037897321832od_b_c
        @ ( product_Sigma_b_c @ A4
          @ ^ [Uu: b] : B5 )
        @ S )
     => ( ( member_c @ Y4 @ B5 )
       => ( ord_less_eq_set_b @ A4 @ ( image_7762865866877615260_b_c_b @ product_fst_b_c @ S ) ) ) ) ).

% subset_fst_imageI
thf(fact_1242_subset__fst__imageI,axiom,
    ! [A4: set_b,B5: set_op6398572775360143365_c_nat,S: set_Pr8806432033423503795_c_nat,Y4: option7520157102916957007_c_nat] :
      ( ( ord_le7634486248364433939_c_nat
        @ ( produc1892124643397453190_c_nat @ A4
          @ ^ [Uu: b] : B5 )
        @ S )
     => ( ( member3208607818295432806_c_nat @ Y4 @ B5 )
       => ( ord_less_eq_set_b @ A4 @ ( image_2722527542522727419_nat_b @ produc8816143915943518219_c_nat @ S ) ) ) ) ).

% subset_fst_imageI
thf(fact_1243_pred__subset__eq,axiom,
    ! [R: set_b,S: set_b] :
      ( ( ord_less_eq_b_o
        @ ^ [X3: b] : ( member_b @ X3 @ R )
        @ ^ [X3: b] : ( member_b @ X3 @ S ) )
      = ( ord_less_eq_set_b @ R @ S ) ) ).

% pred_subset_eq
thf(fact_1244_less__eq__set__def,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A6: set_b,B6: set_b] :
          ( ord_less_eq_b_o
          @ ^ [X3: b] : ( member_b @ X3 @ A6 )
          @ ^ [X3: b] : ( member_b @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_1245_pred__subset__eq2,axiom,
    ! [R: set_Pr8806432033423503795_c_nat,S: set_Pr8806432033423503795_c_nat] :
      ( ( ord_le4248069829227848518_nat_o
        @ ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X3 @ Y3 ) @ R )
        @ ^ [X3: b,Y3: option7520157102916957007_c_nat] : ( member7562873241046315796_c_nat @ ( produc5716802255202478839_c_nat @ X3 @ Y3 ) @ S ) )
      = ( ord_le7634486248364433939_c_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_1246_pred__subset__eq2,axiom,
    ! [R: set_Pr6903500605879609269_c_nat,S: set_Pr6903500605879609269_c_nat] :
      ( ( ord_less_eq_c_nat_o
        @ ^ [X3: c,Y3: nat] : ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X3 @ Y3 ) @ R )
        @ ^ [X3: c,Y3: nat] : ( member8195077246299207702_c_nat @ ( product_Pair_c_nat @ X3 @ Y3 ) @ S ) )
      = ( ord_le1411700432677832725_c_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_1247_finite__inverse__image__gen,axiom,
    ! [A4: set_b,F: b > b,D2: set_b] :
      ( ( finite_finite_b @ A4 )
     => ( ( inj_on_b_b @ F @ D2 )
       => ( finite_finite_b
          @ ( collect_b
            @ ^ [J: b] :
                ( ( member_b @ J @ D2 )
                & ( member_b @ ( F @ J ) @ A4 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_1248_finite__Map__induct,axiom,
    ! [M: b > option2970150418924381261_c_nat,P2: ( b > option2970150418924381261_c_nat ) > $o] :
      ( ( finite_finite_b @ ( dom_b_2330607713053947690_c_nat @ M ) )
     => ( ( P2
          @ ^ [X3: b] : none_P8487441334512977628_c_nat )
       => ( ! [K3: b,V3: produc4862256710654508797_c_nat,M3: b > option2970150418924381261_c_nat] :
              ( ( finite_finite_b @ ( dom_b_2330607713053947690_c_nat @ M3 ) )
             => ( ~ ( member_b @ K3 @ ( dom_b_2330607713053947690_c_nat @ M3 ) )
               => ( ( P2 @ M3 )
                 => ( P2 @ ( fun_up6679455600583992207_c_nat @ M3 @ K3 @ ( some_P2720002978653898840_c_nat @ V3 ) ) ) ) ) )
         => ( P2 @ M ) ) ) ) ).

% finite_Map_induct
thf(fact_1249_max__ext_Omax__extI,axiom,
    ! [X7: set_b,Y8: set_b,R: set_Product_prod_b_b] :
      ( ( finite_finite_b @ X7 )
     => ( ( finite_finite_b @ Y8 )
       => ( ( Y8 != bot_bot_set_b )
         => ( ! [X: b] :
                ( ( member_b @ X @ X7 )
               => ? [Xa3: b] :
                    ( ( member_b @ Xa3 @ Y8 )
                    & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X @ Xa3 ) @ R ) ) )
           => ( member318967379524898064_set_b @ ( produc1352782758248380759_set_b @ X7 @ Y8 ) @ ( max_ext_b @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1250_max__ext_Osimps,axiom,
    ! [A1: set_b,A22: set_b,R: set_Product_prod_b_b] :
      ( ( member318967379524898064_set_b @ ( produc1352782758248380759_set_b @ A1 @ A22 ) @ ( max_ext_b @ R ) )
      = ( ( finite_finite_b @ A1 )
        & ( finite_finite_b @ A22 )
        & ( A22 != bot_bot_set_b )
        & ! [X3: b] :
            ( ( member_b @ X3 @ A1 )
           => ? [Y3: b] :
                ( ( member_b @ Y3 @ A22 )
                & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_1251_max__ext_Ocases,axiom,
    ! [A1: set_b,A22: set_b,R: set_Product_prod_b_b] :
      ( ( member318967379524898064_set_b @ ( produc1352782758248380759_set_b @ A1 @ A22 ) @ ( max_ext_b @ R ) )
     => ~ ( ( finite_finite_b @ A1 )
         => ( ( finite_finite_b @ A22 )
           => ( ( A22 != bot_bot_set_b )
             => ~ ! [X9: b] :
                    ( ( member_b @ X9 @ A1 )
                   => ? [Xa2: b] :
                        ( ( member_b @ Xa2 @ A22 )
                        & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X9 @ Xa2 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1252_insert__relcomp__union__fold,axiom,
    ! [S: set_Pr1964768744408946317_c_nat,X4: produc6689720525226317995_c_nat,X7: set_Pr8806432033423503795_c_nat] :
      ( ( finite2531113271209645270_c_nat @ S )
     => ( ( sup_su1518086237216066783_c_nat @ ( relcom2873162752072038412_c_nat @ ( insert5634363330804326107_c_nat @ X4 @ bot_bo1267386999437214197_c_nat ) @ S ) @ X7 )
        = ( finite475712411486354712_c_nat
          @ ( produc6887334413368000895_c_nat
            @ ^ [W: produc4862256710654508797_c_nat,Z4: option7520157102916957007_c_nat,A8: set_Pr8806432033423503795_c_nat] :
                ( if_set3470817297484139629_c_nat
                @ ( ( produc6319817161620345591_c_nat @ X4 )
                  = W )
                @ ( insert8034245793825879341_c_nat @ ( produc5716802255202478839_c_nat @ ( produc6033490199168946105_c_nat @ X4 ) @ Z4 ) @ A8 )
                @ A8 ) )
          @ X7
          @ S ) ) ) ).

% insert_relcomp_union_fold
thf(fact_1253_insert__relcomp__union__fold,axiom,
    ! [S: set_Pr6411926689957953204_c_nat,X4: product_prod_b_c,X7: set_Pr8806432033423503795_c_nat] :
      ( ( finite3333252062135059861_c_nat @ S )
     => ( ( sup_su1518086237216066783_c_nat @ ( relcom6857592940063405457_c_nat @ ( insert1747480808674937912od_b_c @ X4 @ bot_bo2863795366942399540od_b_c ) @ S ) @ X7 )
        = ( finite2867452577164201723_c_nat
          @ ( produc5491469990890017450_c_nat
            @ ^ [W: c,Z4: option7520157102916957007_c_nat,A8: set_Pr8806432033423503795_c_nat] :
                ( if_set3470817297484139629_c_nat
                @ ( ( product_snd_b_c @ X4 )
                  = W )
                @ ( insert8034245793825879341_c_nat @ ( produc5716802255202478839_c_nat @ ( product_fst_b_c @ X4 ) @ Z4 ) @ A8 )
                @ A8 ) )
          @ X7
          @ S ) ) ) ).

% insert_relcomp_union_fold
thf(fact_1254_insert__relcomp__union__fold,axiom,
    ! [S: set_Pr4914646408830571231_c_nat,X4: produc4862256710654508797_c_nat,X7: set_Pr8806432033423503795_c_nat] :
      ( ( finite4878842257908346408_c_nat @ S )
     => ( ( sup_su1518086237216066783_c_nat @ ( relcom177692130347456094_c_nat @ ( insert8034245793825879341_c_nat @ X4 @ bot_bo7474904860610844231_c_nat ) @ S ) @ X7 )
        = ( finite6725482468114840938_c_nat
          @ ( produc8838197355213761489_c_nat
            @ ^ [W: option7520157102916957007_c_nat,Z4: option7520157102916957007_c_nat,A8: set_Pr8806432033423503795_c_nat] :
                ( if_set3470817297484139629_c_nat
                @ ( ( produc2435110264752509257_c_nat @ X4 )
                  = W )
                @ ( insert8034245793825879341_c_nat @ ( produc5716802255202478839_c_nat @ ( produc8816143915943518219_c_nat @ X4 ) @ Z4 ) @ A8 )
                @ A8 ) )
          @ X7
          @ S ) ) ) ).

% insert_relcomp_union_fold
thf(fact_1255_sorted__wrt__map,axiom,
    ! [R: b > b > $o,F: produc6689720525226317995_c_nat > b,Xs: list_P7417839048565863355_c_nat] :
      ( ( sorted_wrt_b @ R @ ( map_Pr6200325787298368846_nat_b @ F @ Xs ) )
      = ( sorted5187701906718811902_c_nat
        @ ^ [X3: produc6689720525226317995_c_nat,Y3: produc6689720525226317995_c_nat] : ( R @ ( F @ X3 ) @ ( F @ Y3 ) )
        @ Xs ) ) ).

% sorted_wrt_map
thf(fact_1256_sorted__wrt__map,axiom,
    ! [R: b > b > $o,F: product_prod_b_c > b,Xs: list_P903359562653991662od_b_c] :
      ( ( sorted_wrt_b @ R @ ( map_Pr8222292247188198875_b_c_b @ F @ Xs ) )
      = ( sorted4233907796234092629od_b_c
        @ ^ [X3: product_prod_b_c,Y3: product_prod_b_c] : ( R @ ( F @ X3 ) @ ( F @ Y3 ) )
        @ Xs ) ) ).

% sorted_wrt_map
thf(fact_1257_product__fold,axiom,
    ! [A4: set_b,B5: set_op6398572775360143365_c_nat] :
      ( ( finite_finite_b @ A4 )
     => ( ( finite1864547194484239206_c_nat @ B5 )
       => ( ( produc1892124643397453190_c_nat @ A4
            @ ^ [Uu: b] : B5 )
          = ( finite1684880752614361450_c_nat
            @ ^ [X3: b,Z4: set_Pr8806432033423503795_c_nat] :
                ( finite5055797010418863948_c_nat
                @ ^ [Y3: option7520157102916957007_c_nat] : ( insert8034245793825879341_c_nat @ ( produc5716802255202478839_c_nat @ X3 @ Y3 ) )
                @ Z4
                @ B5 )
            @ bot_bo7474904860610844231_c_nat
            @ A4 ) ) ) ) ).

% product_fold
thf(fact_1258_product__fold,axiom,
    ! [A4: set_c,B5: set_nat] :
      ( ( finite_finite_c @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( product_Sigma_c_nat @ A4
            @ ^ [Uu: c] : B5 )
          = ( finite3928500458190289901_c_nat
            @ ^ [X3: c,Z4: set_Pr6903500605879609269_c_nat] :
                ( finite7245807490894926287_c_nat
                @ ^ [Y3: nat] : ( insert1302120044651762863_c_nat @ ( product_Pair_c_nat @ X3 @ Y3 ) )
                @ Z4
                @ B5 )
            @ bot_bo1794802125927853641_c_nat
            @ A4 ) ) ) ) ).

% product_fold
thf(fact_1259_insert__relcomp__fold,axiom,
    ! [S: set_Pr1964768744408946317_c_nat,X4: produc6689720525226317995_c_nat,R: set_Pr93524137074774369_c_nat] :
      ( ( finite2531113271209645270_c_nat @ S )
     => ( ( relcom2873162752072038412_c_nat @ ( insert5634363330804326107_c_nat @ X4 @ R ) @ S )
        = ( finite475712411486354712_c_nat
          @ ( produc6887334413368000895_c_nat
            @ ^ [W: produc4862256710654508797_c_nat,Z4: option7520157102916957007_c_nat,A8: set_Pr8806432033423503795_c_nat] :
                ( if_set3470817297484139629_c_nat
                @ ( ( produc6319817161620345591_c_nat @ X4 )
                  = W )
                @ ( insert8034245793825879341_c_nat @ ( produc5716802255202478839_c_nat @ ( produc6033490199168946105_c_nat @ X4 ) @ Z4 ) @ A8 )
                @ A8 ) )
          @ ( relcom2873162752072038412_c_nat @ R @ S )
          @ S ) ) ) ).

% insert_relcomp_fold
thf(fact_1260_insert__relcomp__fold,axiom,
    ! [S: set_Pr6411926689957953204_c_nat,X4: product_prod_b_c,R: set_Product_prod_b_c] :
      ( ( finite3333252062135059861_c_nat @ S )
     => ( ( relcom6857592940063405457_c_nat @ ( insert1747480808674937912od_b_c @ X4 @ R ) @ S )
        = ( finite2867452577164201723_c_nat
          @ ( produc5491469990890017450_c_nat
            @ ^ [W: c,Z4: option7520157102916957007_c_nat,A8: set_Pr8806432033423503795_c_nat] :
                ( if_set3470817297484139629_c_nat
                @ ( ( product_snd_b_c @ X4 )
                  = W )
                @ ( insert8034245793825879341_c_nat @ ( produc5716802255202478839_c_nat @ ( product_fst_b_c @ X4 ) @ Z4 ) @ A8 )
                @ A8 ) )
          @ ( relcom6857592940063405457_c_nat @ R @ S )
          @ S ) ) ) ).

% insert_relcomp_fold
thf(fact_1261_insert__relcomp__fold,axiom,
    ! [S: set_Pr4914646408830571231_c_nat,X4: produc4862256710654508797_c_nat,R: set_Pr8806432033423503795_c_nat] :
      ( ( finite4878842257908346408_c_nat @ S )
     => ( ( relcom177692130347456094_c_nat @ ( insert8034245793825879341_c_nat @ X4 @ R ) @ S )
        = ( finite6725482468114840938_c_nat
          @ ( produc8838197355213761489_c_nat
            @ ^ [W: option7520157102916957007_c_nat,Z4: option7520157102916957007_c_nat,A8: set_Pr8806432033423503795_c_nat] :
                ( if_set3470817297484139629_c_nat
                @ ( ( produc2435110264752509257_c_nat @ X4 )
                  = W )
                @ ( insert8034245793825879341_c_nat @ ( produc5716802255202478839_c_nat @ ( produc8816143915943518219_c_nat @ X4 ) @ Z4 ) @ A8 )
                @ A8 ) )
          @ ( relcom177692130347456094_c_nat @ R @ S )
          @ S ) ) ) ).

% insert_relcomp_fold
thf(fact_1262_Set_Ofilter__def,axiom,
    ( filter_b2
    = ( ^ [P4: b > $o,A6: set_b] :
          ( collect_b
          @ ^ [A5: b] :
              ( ( member_b @ A5 @ A6 )
              & ( P4 @ A5 ) ) ) ) ) ).

% Set.filter_def
thf(fact_1263_comp__fun__commute__product__fold,axiom,
    ! [B5: set_op6398572775360143365_c_nat] :
      ( ( finite1864547194484239206_c_nat @ B5 )
     => ( finite3390218106231828257_c_nat
        @ ^ [X3: b,Z4: set_Pr8806432033423503795_c_nat] :
            ( finite5055797010418863948_c_nat
            @ ^ [Y3: option7520157102916957007_c_nat] : ( insert8034245793825879341_c_nat @ ( produc5716802255202478839_c_nat @ X3 @ Y3 ) )
            @ Z4
            @ B5 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_1264_comp__fun__commute__product__fold,axiom,
    ! [B5: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( finite8457442031682293540_c_nat
        @ ^ [X3: c,Z4: set_Pr6903500605879609269_c_nat] :
            ( finite7245807490894926287_c_nat
            @ ^ [Y3: nat] : ( insert1302120044651762863_c_nat @ ( product_Pair_c_nat @ X3 @ Y3 ) )
            @ Z4
            @ B5 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_1265_fold__union__pair,axiom,
    ! [B5: set_op6398572775360143365_c_nat,X4: b,A4: set_Pr8806432033423503795_c_nat] :
      ( ( finite1864547194484239206_c_nat @ B5 )
     => ( ( sup_su1518086237216066783_c_nat
          @ ( comple7028538466571265158_c_nat
            @ ( image_3246528641730782985_c_nat
              @ ^ [Y3: option7520157102916957007_c_nat] : ( insert8034245793825879341_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y3 ) @ bot_bo7474904860610844231_c_nat )
              @ B5 ) )
          @ A4 )
        = ( finite5055797010418863948_c_nat
          @ ^ [Y3: option7520157102916957007_c_nat] : ( insert8034245793825879341_c_nat @ ( produc5716802255202478839_c_nat @ X4 @ Y3 ) )
          @ A4
          @ B5 ) ) ) ).

% fold_union_pair
thf(fact_1266_fold__union__pair,axiom,
    ! [B5: set_nat,X4: c,A4: set_Pr6903500605879609269_c_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( sup_su2428977078917480673_c_nat
          @ ( comple5740988929479974408_c_nat
            @ ( image_7176720838824768140_c_nat
              @ ^ [Y3: nat] : ( insert1302120044651762863_c_nat @ ( product_Pair_c_nat @ X4 @ Y3 ) @ bot_bo1794802125927853641_c_nat )
              @ B5 ) )
          @ A4 )
        = ( finite7245807490894926287_c_nat
          @ ^ [Y3: nat] : ( insert1302120044651762863_c_nat @ ( product_Pair_c_nat @ X4 @ Y3 ) )
          @ A4
          @ B5 ) ) ) ).

% fold_union_pair
thf(fact_1267_UN__I,axiom,
    ! [A: b,A4: set_b,B: b,B5: b > set_b] :
      ( ( member_b @ A @ A4 )
     => ( ( member_b @ B @ ( B5 @ A ) )
       => ( member_b @ B @ ( comple2307003614231284044_set_b @ ( image_b_set_b @ B5 @ A4 ) ) ) ) ) ).

% UN_I
thf(fact_1268_UN__E,axiom,
    ! [B: b,B5: b > set_b,A4: set_b] :
      ( ( member_b @ B @ ( comple2307003614231284044_set_b @ ( image_b_set_b @ B5 @ A4 ) ) )
     => ~ ! [X: b] :
            ( ( member_b @ X @ A4 )
           => ~ ( member_b @ B @ ( B5 @ X ) ) ) ) ).

% UN_E

% Helper facts (17)
thf(help_If_2_1_If_001t__Option__Ooption_Itf__b_J_T,axiom,
    ! [X4: option_b,Y4: option_b] :
      ( ( if_option_b @ $false @ X4 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_Itf__b_J_T,axiom,
    ! [X4: option_b,Y4: option_b] :
      ( ( if_option_b @ $true @ X4 @ Y4 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_Itf__c_J_T,axiom,
    ! [X4: option_c,Y4: option_c] :
      ( ( if_option_c @ $false @ X4 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_Itf__c_J_T,axiom,
    ! [X4: option_c,Y4: option_c] :
      ( ( if_option_c @ $true @ X4 @ Y4 )
      = X4 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_T,axiom,
    ! [X4: list_P5561950507774946575_c_nat,Y4: list_P5561950507774946575_c_nat] :
      ( ( if_lis8000280747373556681_c_nat @ $false @ X4 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_T,axiom,
    ! [X4: list_P5561950507774946575_c_nat,Y4: list_P5561950507774946575_c_nat] :
      ( ( if_lis8000280747373556681_c_nat @ $true @ X4 @ Y4 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_T,axiom,
    ! [X4: option7520157102916957007_c_nat,Y4: option7520157102916957007_c_nat] :
      ( ( if_opt8655011569862983689_c_nat @ $false @ X4 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_T,axiom,
    ! [X4: option7520157102916957007_c_nat,Y4: option7520157102916957007_c_nat] :
      ( ( if_opt8655011569862983689_c_nat @ $true @ X4 @ Y4 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X4: produc4862256710654508797_c_nat,Y4: produc4862256710654508797_c_nat] :
      ( ( if_Pro7338708846875558327_c_nat @ $false @ X4 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X4: produc4862256710654508797_c_nat,Y4: produc4862256710654508797_c_nat] :
      ( ( if_Pro7338708846875558327_c_nat @ $true @ X4 @ Y4 )
      = X4 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X4: set_Pr8806432033423503795_c_nat,Y4: set_Pr8806432033423503795_c_nat] :
      ( ( if_set3470817297484139629_c_nat @ $false @ X4 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X4: set_Pr8806432033423503795_c_nat,Y4: set_Pr8806432033423503795_c_nat] :
      ( ( if_set3470817297484139629_c_nat @ $true @ X4 @ Y4 )
      = X4 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X4: list_P8155366108056616461_c_nat,Y4: list_P8155366108056616461_c_nat] :
      ( ( if_lis6091463167075048391_c_nat @ $false @ X4 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X4: list_P8155366108056616461_c_nat,Y4: list_P8155366108056616461_c_nat] :
      ( ( if_lis6091463167075048391_c_nat @ $true @ X4 @ Y4 )
      = X4 ) ).

thf(help_If_3_1_If_001t__Option__Ooption_It__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X4: option2970150418924381261_c_nat,Y4: option2970150418924381261_c_nat] :
      ( ( if_opt6485458775533684743_c_nat @ $false @ X4 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Product____Type__Oprod_Itf__b_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X4: option2970150418924381261_c_nat,Y4: option2970150418924381261_c_nat] :
      ( ( if_opt6485458775533684743_c_nat @ $true @ X4 @ Y4 )
      = X4 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( mmap_u2662500248198004552_c_nat @ init
      @ ( case_o1602653449097480839_c_nat @ ( produc5716802255202478839_c_nat @ q_cur @ tstp_cur )
        @ ( produc3722540595886809633_c_nat
          @ ^ [Q: b,Tstp: option7520157102916957007_c_nat] :
              ( case_o6498799718091186953_c_nat @ ( produc5716802255202478839_c_nat @ Q @ tstp_cur )
              @ ( produc3293230613241633187_c_nat
                @ ^ [Ts: c,Tp: nat] : ( produc5716802255202478839_c_nat @ Q @ Tstp ) )
              @ Tstp ) )
        @ ( mmap_l5026999719965937591_c_nat @ s_cur @ q_cur ) )
      @ s2 )
    = ( mmap_u2662500248198004552_c_nat @ init @ ( produc5716802255202478839_c_nat @ q_cur @ tstp_cur ) @ s2 ) ) ).

%------------------------------------------------------------------------------