%------------------------------------------------------------------------------
% 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_01292_056555__16449706_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2323 ( 777 unt;1022 typ;   0 def)
%            Number of atoms       : 3167 (1938 equ;   0 cnn)
%            Maximal formula atoms :   14 (   2 avg)
%            Number of connectives : 12615 ( 630   ~;  58   |; 258   &;10327   @)
%                                         (   0 <=>;1342  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   6 avg)
%            Number of types       :  197 ( 196 usr)
%            Number of type conns  : 3563 (3563   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  829 ( 826 usr;  50 con; 0-5 aty)
%            Number of variables   : 5192 ( 955   ^;4053   !; 184   ?;5192   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:53:16.402
%------------------------------------------------------------------------------
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% Explicit typings (826)
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thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_Mtf__a_J_J,type,
    member7689197028375331655_b_a_a: produc3619439354540788382_b_a_a > set_Pr7659163277375235966_b_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J,type,
    member3694737009695996203_a_b_a: produc7822699449245016212_a_b_a > set_Pr2084289749507318346_a_b_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J,type,
    member6046481508403577027_d_nat: produc1990351791824763820_d_nat > set_Pr8583615357880026082_d_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    member1426531481828664017od_a_b: product_prod_a_b > set_Product_prod_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__a_J,type,
    member7862447932407534991od_b_a: product_prod_b_a > set_Product_prod_b_a > $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__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J,type,
    member8156885303890938774_d_nat: produc5456268773499131775_d_nat > set_Pr7704721104634554677_d_nat > $o ).

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

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

thf(sy_v_bs,type,
    bs: b ).

thf(sy_v_i,type,
    i: nat ).

thf(sy_v_inv____,type,
    inv: mappin2365780136655028071_a_b_a > $o ).

thf(sy_v_r,type,
    r: list_P6684558232516339260_d_nat ).

thf(sy_v_s,type,
    s: list_P6684558232516339260_d_nat ).

thf(sy_v_s_H,type,
    s2: list_P6684558232516339260_d_nat ).

thf(sy_v_st,type,
    st: mappin2365780136655028071_a_b_a ).

thf(sy_v_st_H,type,
    st2: mappin2365780136655028071_a_b_a ).

thf(sy_v_step,type,
    step: a > b > a ).

% Relevant facts (1265)
thf(fact_0_inv,axiom,
    ! [Q: a,Bs: b] :
      ( case_option_o_a @ $true
      @ ( ^ [Y: a,Z: a] : ( Y = Z )
        @ ( step @ Q @ Bs ) )
      @ ( lookup863726980359885586_a_b_a @ st @ ( product_Pair_a_b @ Q @ Bs ) ) ) ).

% inv
thf(fact_1_case__prod__conv,axiom,
    ! [F: a > produc5456268773499131775_d_nat > produc1990351791824763820_d_nat,A: a,B: produc5456268773499131775_d_nat] :
      ( ( produc5272039204407661311_d_nat @ F @ ( produc5865182755557626534_d_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_2_case__prod__conv,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,A: a,B: mappin2365780136655028071_a_b_a] :
      ( ( produc452744236373541113_a_b_a @ F @ ( produc1583556749610718734_a_b_a @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_3_case__prod__conv,axiom,
    ! [F: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a] :
      ( ( produc5613835436409813491_a_b_a @ F @ ( produc1681437058057156268_a_b_a @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_4_case__prod__conv,axiom,
    ! [F: a > produc5456268773499131775_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,A: a,B: produc5456268773499131775_d_nat] :
      ( ( produc5638855421663032305_a_b_a @ F @ ( produc5865182755557626534_d_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5_case__prod__conv,axiom,
    ! [F: a > b > product_prod_a_b,A: a,B: b] :
      ( ( produc8992199381948149691od_a_b @ F @ ( product_Pair_a_b @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_6_case__prod__conv,axiom,
    ! [F: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > produc2062766750192416900_a_b_a,A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a] :
      ( ( produc3256888519411912691_a_b_a @ F @ ( produc2616710729536263484_a_b_a @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_7_case__prod__conv,axiom,
    ! [F: a > b > $o,A: a,B: b] :
      ( ( produc3537405659489547051_a_b_o @ F @ ( product_Pair_a_b @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_8_case__prod__conv,axiom,
    ! [F: a > b > list_P1467974523212632366od_a_b,A: a,B: b] :
      ( ( produc611504906096598721od_a_b @ F @ ( product_Pair_a_b @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_9_case__prod__conv,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > $o,A: a,B: mappin2365780136655028071_a_b_a] :
      ( ( produc2053338206788791453_b_a_o @ F @ ( produc1583556749610718734_a_b_a @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_10_case__prod__conv,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > produc7822699449245016212_a_b_a,A: a,B: mappin2365780136655028071_a_b_a] :
      ( ( produc543687385667932879_a_b_a @ F @ ( produc1583556749610718734_a_b_a @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_11_disjE__realizer2,axiom,
    ! [P: $o,Q2: product_prod_d_nat > $o,X: option3893003681250311888_d_nat,R: option3893003681250311888_d_nat > $o,F: option3893003681250311888_d_nat,G: product_prod_d_nat > option3893003681250311888_d_nat] :
      ( ( case_o4860870191544392141_d_nat @ P @ Q2 @ X )
     => ( ( P
         => ( R @ F ) )
       => ( ! [Q3: product_prod_d_nat] :
              ( ( Q2 @ Q3 )
             => ( R @ ( G @ Q3 ) ) )
         => ( R @ ( case_o6579539980594861021_d_nat @ F @ G @ X ) ) ) ) ) ).

% disjE_realizer2
thf(fact_12_disjE__realizer2,axiom,
    ! [P: $o,Q2: a > $o,X: option_a,R: produc7822699449245016212_a_b_a > $o,F: produc7822699449245016212_a_b_a,G: a > produc7822699449245016212_a_b_a] :
      ( ( case_option_o_a @ P @ Q2 @ X )
     => ( ( P
         => ( R @ F ) )
       => ( ! [Q3: a] :
              ( ( Q2 @ Q3 )
             => ( R @ ( G @ Q3 ) ) )
         => ( R @ ( case_o140846598303662877_b_a_a @ F @ G @ X ) ) ) ) ) ).

% disjE_realizer2
thf(fact_13_disjE__realizer2,axiom,
    ! [P: $o,Q2: a > $o,X: option_a,R: mappin2365780136655028071_a_b_a > $o,F: mappin2365780136655028071_a_b_a,G: a > mappin2365780136655028071_a_b_a] :
      ( ( case_option_o_a @ P @ Q2 @ X )
     => ( ( P
         => ( R @ F ) )
       => ( ! [Q3: a] :
              ( ( Q2 @ Q3 )
             => ( R @ ( G @ Q3 ) ) )
         => ( R @ ( case_o2154183385986904842_b_a_a @ F @ G @ X ) ) ) ) ) ).

% disjE_realizer2
thf(fact_14_disjE__realizer2,axiom,
    ! [P: $o,Q2: a > $o,X: option_a,R: option_a > $o,F: option_a,G: a > option_a] :
      ( ( case_option_o_a @ P @ Q2 @ X )
     => ( ( P
         => ( R @ F ) )
       => ( ! [Q3: a] :
              ( ( Q2 @ Q3 )
             => ( R @ ( G @ Q3 ) ) )
         => ( R @ ( case_o3148979394504432965on_a_a @ F @ G @ X ) ) ) ) ) ).

% disjE_realizer2
thf(fact_15_disjE__realizer2,axiom,
    ! [P: $o,Q2: a > $o,X: option_a,R: a > $o,F: a,G: a > a] :
      ( ( case_option_o_a @ P @ Q2 @ X )
     => ( ( P
         => ( R @ F ) )
       => ( ! [Q3: a] :
              ( ( Q2 @ Q3 )
             => ( R @ ( G @ Q3 ) ) )
         => ( R @ ( case_option_a_a @ F @ G @ X ) ) ) ) ) ).

% disjE_realizer2
thf(fact_16_disjE__realizer2,axiom,
    ! [P: $o,Q2: a > $o,X: option_a,R: $o > $o,F: $o,G: a > $o] :
      ( ( case_option_o_a @ P @ Q2 @ X )
     => ( ( P
         => ( R @ F ) )
       => ( ! [Q3: a] :
              ( ( Q2 @ Q3 )
             => ( R @ ( G @ Q3 ) ) )
         => ( R @ ( case_option_o_a @ F @ G @ X ) ) ) ) ) ).

% disjE_realizer2
thf(fact_17_case__prod__Pair__iden,axiom,
    ! [P2: produc3619439354540788382_b_a_a] :
      ( ( produc1379674575869628219_b_a_a @ produc4259011374121840526_b_a_a @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_18_case__prod__Pair__iden,axiom,
    ! [P2: produc9176665399011442921_d_nat] :
      ( ( produc610653850605767673_d_nat @ produc919746265332163299_d_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_19_case__prod__Pair__iden,axiom,
    ! [P2: produc7083967959564351825_a_b_a] :
      ( ( produc2041615158718508361_a_b_a @ produc6994869176599099851_a_b_a @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_20_case__prod__Pair__iden,axiom,
    ! [P2: produc2639161875728825943od_a_b] :
      ( ( produc5323732787154529659od_a_b @ produc6643264353547226055od_a_b @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_21_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_a_a] :
      ( ( produc408267641121961211od_a_a @ product_Pair_a_a @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_22_case__prod__Pair__iden,axiom,
    ! [P2: produc7822699449245016212_a_b_a] :
      ( ( produc543687385667932879_a_b_a @ produc1583556749610718734_a_b_a @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_23_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_a_b] :
      ( ( produc8992199381948149691od_a_b @ product_Pair_a_b @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_24_case__prod__Pair__iden,axiom,
    ! [P2: produc2062766750192416900_a_b_a] :
      ( ( produc3256888519411912691_a_b_a @ produc2616710729536263484_a_b_a @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_25_case__prod__Pair__iden,axiom,
    ! [P2: produc1990351791824763820_d_nat] :
      ( ( produc5272039204407661311_d_nat @ produc5865182755557626534_d_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_26_case__prod__Pair__iden,axiom,
    ! [P2: produc6343861673719254644_a_b_a] :
      ( ( produc5613835436409813491_a_b_a @ produc1681437058057156268_a_b_a @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_27_prod_Oinject,axiom,
    ! [X1: a,X2: mappin4996136453398522101_b_a_a,Y1: a,Y2: mappin4996136453398522101_b_a_a] :
      ( ( ( produc4259011374121840526_b_a_a @ X1 @ X2 )
        = ( produc4259011374121840526_b_a_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_28_prod_Oinject,axiom,
    ! [X1: a,X2: list_P6684558232516339260_d_nat,Y1: a,Y2: list_P6684558232516339260_d_nat] :
      ( ( ( produc919746265332163299_d_nat @ X1 @ X2 )
        = ( produc919746265332163299_d_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_29_prod_Oinject,axiom,
    ! [X1: a,X2: list_P2992919015626629540_a_b_a,Y1: a,Y2: list_P2992919015626629540_a_b_a] :
      ( ( ( produc6994869176599099851_a_b_a @ X1 @ X2 )
        = ( produc6994869176599099851_a_b_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_30_prod_Oinject,axiom,
    ! [X1: a,X2: list_P1467974523212632366od_a_b,Y1: a,Y2: list_P1467974523212632366od_a_b] :
      ( ( ( produc6643264353547226055od_a_b @ X1 @ X2 )
        = ( produc6643264353547226055od_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_31_prod_Oinject,axiom,
    ! [X1: a,X2: a,Y1: a,Y2: a] :
      ( ( ( product_Pair_a_a @ X1 @ X2 )
        = ( product_Pair_a_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_32_prod_Oinject,axiom,
    ! [X1: a,X2: mappin2365780136655028071_a_b_a,Y1: a,Y2: mappin2365780136655028071_a_b_a] :
      ( ( ( produc1583556749610718734_a_b_a @ X1 @ X2 )
        = ( produc1583556749610718734_a_b_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_33_prod_Oinject,axiom,
    ! [X1: a,X2: produc5456268773499131775_d_nat,Y1: a,Y2: produc5456268773499131775_d_nat] :
      ( ( ( produc5865182755557626534_d_nat @ X1 @ X2 )
        = ( produc5865182755557626534_d_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_34_prod_Oinject,axiom,
    ! [X1: a,X2: b,Y1: a,Y2: b] :
      ( ( ( product_Pair_a_b @ X1 @ X2 )
        = ( product_Pair_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_35_prod_Oinject,axiom,
    ! [X1: produc1990351791824763820_d_nat,X2: mappin2365780136655028071_a_b_a,Y1: produc1990351791824763820_d_nat,Y2: mappin2365780136655028071_a_b_a] :
      ( ( ( produc1681437058057156268_a_b_a @ X1 @ X2 )
        = ( produc1681437058057156268_a_b_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_36_prod_Oinject,axiom,
    ! [X1: list_P6684558232516339260_d_nat,X2: mappin2365780136655028071_a_b_a,Y1: list_P6684558232516339260_d_nat,Y2: mappin2365780136655028071_a_b_a] :
      ( ( ( produc2616710729536263484_a_b_a @ X1 @ X2 )
        = ( produc2616710729536263484_a_b_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_37_old_Oprod_Oinject,axiom,
    ! [A: a,B: mappin4996136453398522101_b_a_a,A2: a,B2: mappin4996136453398522101_b_a_a] :
      ( ( ( produc4259011374121840526_b_a_a @ A @ B )
        = ( produc4259011374121840526_b_a_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_38_old_Oprod_Oinject,axiom,
    ! [A: a,B: list_P6684558232516339260_d_nat,A2: a,B2: list_P6684558232516339260_d_nat] :
      ( ( ( produc919746265332163299_d_nat @ A @ B )
        = ( produc919746265332163299_d_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_39_old_Oprod_Oinject,axiom,
    ! [A: a,B: list_P2992919015626629540_a_b_a,A2: a,B2: list_P2992919015626629540_a_b_a] :
      ( ( ( produc6994869176599099851_a_b_a @ A @ B )
        = ( produc6994869176599099851_a_b_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_40_old_Oprod_Oinject,axiom,
    ! [A: a,B: list_P1467974523212632366od_a_b,A2: a,B2: list_P1467974523212632366od_a_b] :
      ( ( ( produc6643264353547226055od_a_b @ A @ B )
        = ( produc6643264353547226055od_a_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_41_old_Oprod_Oinject,axiom,
    ! [A: a,B: a,A2: a,B2: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_42_old_Oprod_Oinject,axiom,
    ! [A: a,B: mappin2365780136655028071_a_b_a,A2: a,B2: mappin2365780136655028071_a_b_a] :
      ( ( ( produc1583556749610718734_a_b_a @ A @ B )
        = ( produc1583556749610718734_a_b_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_43_old_Oprod_Oinject,axiom,
    ! [A: a,B: produc5456268773499131775_d_nat,A2: a,B2: produc5456268773499131775_d_nat] :
      ( ( ( produc5865182755557626534_d_nat @ A @ B )
        = ( produc5865182755557626534_d_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_44_old_Oprod_Oinject,axiom,
    ! [A: a,B: b,A2: a,B2: b] :
      ( ( ( product_Pair_a_b @ A @ B )
        = ( product_Pair_a_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_45_old_Oprod_Oinject,axiom,
    ! [A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a,A2: produc1990351791824763820_d_nat,B2: mappin2365780136655028071_a_b_a] :
      ( ( ( produc1681437058057156268_a_b_a @ A @ B )
        = ( produc1681437058057156268_a_b_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_46_old_Oprod_Oinject,axiom,
    ! [A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a,A2: list_P6684558232516339260_d_nat,B2: mappin2365780136655028071_a_b_a] :
      ( ( ( produc2616710729536263484_a_b_a @ A @ B )
        = ( produc2616710729536263484_a_b_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_47_case__prodE2,axiom,
    ! [Q2: produc1990351791824763820_d_nat > $o,P: a > produc5456268773499131775_d_nat > produc1990351791824763820_d_nat,Z2: produc1990351791824763820_d_nat] :
      ( ( Q2 @ ( produc5272039204407661311_d_nat @ P @ Z2 ) )
     => ~ ! [X3: a,Y3: produc5456268773499131775_d_nat] :
            ( ( Z2
              = ( produc5865182755557626534_d_nat @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_48_case__prodE2,axiom,
    ! [Q2: produc6343861673719254644_a_b_a > $o,P: a > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,Z2: produc7822699449245016212_a_b_a] :
      ( ( Q2 @ ( produc452744236373541113_a_b_a @ P @ Z2 ) )
     => ~ ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
            ( ( Z2
              = ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_49_case__prodE2,axiom,
    ! [Q2: produc6343861673719254644_a_b_a > $o,P: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,Z2: produc6343861673719254644_a_b_a] :
      ( ( Q2 @ ( produc5613835436409813491_a_b_a @ P @ Z2 ) )
     => ~ ! [X3: produc1990351791824763820_d_nat,Y3: mappin2365780136655028071_a_b_a] :
            ( ( Z2
              = ( produc1681437058057156268_a_b_a @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_50_case__prodE2,axiom,
    ! [Q2: ( mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a ) > $o,P: a > produc5456268773499131775_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,Z2: produc1990351791824763820_d_nat] :
      ( ( Q2 @ ( produc5638855421663032305_a_b_a @ P @ Z2 ) )
     => ~ ! [X3: a,Y3: produc5456268773499131775_d_nat] :
            ( ( Z2
              = ( produc5865182755557626534_d_nat @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_51_case__prodE2,axiom,
    ! [Q2: product_prod_a_b > $o,P: a > b > product_prod_a_b,Z2: product_prod_a_b] :
      ( ( Q2 @ ( produc8992199381948149691od_a_b @ P @ Z2 ) )
     => ~ ! [X3: a,Y3: b] :
            ( ( Z2
              = ( product_Pair_a_b @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_52_case__prodE2,axiom,
    ! [Q2: produc2062766750192416900_a_b_a > $o,P: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > produc2062766750192416900_a_b_a,Z2: produc2062766750192416900_a_b_a] :
      ( ( Q2 @ ( produc3256888519411912691_a_b_a @ P @ Z2 ) )
     => ~ ! [X3: list_P6684558232516339260_d_nat,Y3: mappin2365780136655028071_a_b_a] :
            ( ( Z2
              = ( produc2616710729536263484_a_b_a @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_53_case__prodE2,axiom,
    ! [Q2: $o > $o,P: a > b > $o,Z2: product_prod_a_b] :
      ( ( Q2 @ ( produc3537405659489547051_a_b_o @ P @ Z2 ) )
     => ~ ! [X3: a,Y3: b] :
            ( ( Z2
              = ( product_Pair_a_b @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_54_case__prodE2,axiom,
    ! [Q2: list_P1467974523212632366od_a_b > $o,P: a > b > list_P1467974523212632366od_a_b,Z2: product_prod_a_b] :
      ( ( Q2 @ ( produc611504906096598721od_a_b @ P @ Z2 ) )
     => ~ ! [X3: a,Y3: b] :
            ( ( Z2
              = ( product_Pair_a_b @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_55_case__prodE2,axiom,
    ! [Q2: $o > $o,P: a > mappin2365780136655028071_a_b_a > $o,Z2: produc7822699449245016212_a_b_a] :
      ( ( Q2 @ ( produc2053338206788791453_b_a_o @ P @ Z2 ) )
     => ~ ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
            ( ( Z2
              = ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_56_case__prodE2,axiom,
    ! [Q2: produc7822699449245016212_a_b_a > $o,P: a > mappin2365780136655028071_a_b_a > produc7822699449245016212_a_b_a,Z2: produc7822699449245016212_a_b_a] :
      ( ( Q2 @ ( produc543687385667932879_a_b_a @ P @ Z2 ) )
     => ~ ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
            ( ( Z2
              = ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_57_case__prod__eta,axiom,
    ! [F: produc1990351791824763820_d_nat > produc1990351791824763820_d_nat] :
      ( ( produc5272039204407661311_d_nat
        @ ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( F @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_58_case__prod__eta,axiom,
    ! [F: produc7822699449245016212_a_b_a > produc6343861673719254644_a_b_a] :
      ( ( produc452744236373541113_a_b_a
        @ ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( F @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_59_case__prod__eta,axiom,
    ! [F: produc6343861673719254644_a_b_a > produc6343861673719254644_a_b_a] :
      ( ( produc5613835436409813491_a_b_a
        @ ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( F @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_60_case__prod__eta,axiom,
    ! [F: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a] :
      ( ( produc5638855421663032305_a_b_a
        @ ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( F @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_61_case__prod__eta,axiom,
    ! [F: product_prod_a_b > product_prod_a_b] :
      ( ( produc8992199381948149691od_a_b
        @ ^ [X4: a,Y4: b] : ( F @ ( product_Pair_a_b @ X4 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_62_case__prod__eta,axiom,
    ! [F: produc2062766750192416900_a_b_a > produc2062766750192416900_a_b_a] :
      ( ( produc3256888519411912691_a_b_a
        @ ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( F @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_63_case__prod__eta,axiom,
    ! [F: product_prod_a_b > $o] :
      ( ( produc3537405659489547051_a_b_o
        @ ^ [X4: a,Y4: b] : ( F @ ( product_Pair_a_b @ X4 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_64_case__prod__eta,axiom,
    ! [F: product_prod_a_b > list_P1467974523212632366od_a_b] :
      ( ( produc611504906096598721od_a_b
        @ ^ [X4: a,Y4: b] : ( F @ ( product_Pair_a_b @ X4 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_65_case__prod__eta,axiom,
    ! [F: produc7822699449245016212_a_b_a > $o] :
      ( ( produc2053338206788791453_b_a_o
        @ ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( F @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_66_case__prod__eta,axiom,
    ! [F: produc7822699449245016212_a_b_a > produc7822699449245016212_a_b_a] :
      ( ( produc543687385667932879_a_b_a
        @ ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( F @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_67_cond__case__prod__eta,axiom,
    ! [F: a > b > product_prod_a_b,G: product_prod_a_b > product_prod_a_b] :
      ( ! [X3: a,Y3: b] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_a_b @ X3 @ Y3 ) ) )
     => ( ( produc8992199381948149691od_a_b @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_68_cond__case__prod__eta,axiom,
    ! [F: a > produc5456268773499131775_d_nat > produc1990351791824763820_d_nat,G: produc1990351791824763820_d_nat > produc1990351791824763820_d_nat] :
      ( ! [X3: a,Y3: produc5456268773499131775_d_nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc5865182755557626534_d_nat @ X3 @ Y3 ) ) )
     => ( ( produc5272039204407661311_d_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_69_cond__case__prod__eta,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,G: produc7822699449245016212_a_b_a > produc6343861673719254644_a_b_a] :
      ( ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) ) )
     => ( ( produc452744236373541113_a_b_a @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_70_cond__case__prod__eta,axiom,
    ! [F: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,G: produc6343861673719254644_a_b_a > produc6343861673719254644_a_b_a] :
      ( ! [X3: produc1990351791824763820_d_nat,Y3: mappin2365780136655028071_a_b_a] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc1681437058057156268_a_b_a @ X3 @ Y3 ) ) )
     => ( ( produc5613835436409813491_a_b_a @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_71_cond__case__prod__eta,axiom,
    ! [F: a > produc5456268773499131775_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,G: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a] :
      ( ! [X3: a,Y3: produc5456268773499131775_d_nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc5865182755557626534_d_nat @ X3 @ Y3 ) ) )
     => ( ( produc5638855421663032305_a_b_a @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_72_cond__case__prod__eta,axiom,
    ! [F: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > produc2062766750192416900_a_b_a,G: produc2062766750192416900_a_b_a > produc2062766750192416900_a_b_a] :
      ( ! [X3: list_P6684558232516339260_d_nat,Y3: mappin2365780136655028071_a_b_a] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc2616710729536263484_a_b_a @ X3 @ Y3 ) ) )
     => ( ( produc3256888519411912691_a_b_a @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_73_cond__case__prod__eta,axiom,
    ! [F: a > b > $o,G: product_prod_a_b > $o] :
      ( ! [X3: a,Y3: b] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_a_b @ X3 @ Y3 ) ) )
     => ( ( produc3537405659489547051_a_b_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_74_cond__case__prod__eta,axiom,
    ! [F: a > b > list_P1467974523212632366od_a_b,G: product_prod_a_b > list_P1467974523212632366od_a_b] :
      ( ! [X3: a,Y3: b] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_a_b @ X3 @ Y3 ) ) )
     => ( ( produc611504906096598721od_a_b @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_75_cond__case__prod__eta,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > $o,G: produc7822699449245016212_a_b_a > $o] :
      ( ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) ) )
     => ( ( produc2053338206788791453_b_a_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_76_cond__case__prod__eta,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > produc7822699449245016212_a_b_a,G: produc7822699449245016212_a_b_a > produc7822699449245016212_a_b_a] :
      ( ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) ) )
     => ( ( produc543687385667932879_a_b_a @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_77_local_Oinv__def,axiom,
    ( inv
    = ( ^ [St: mappin2365780136655028071_a_b_a] :
        ! [Q4: a,Bs2: b] :
          ( case_option_o_a @ $true
          @ ( ^ [Y: a,Z: a] : ( Y = Z )
            @ ( step @ Q4 @ Bs2 ) )
          @ ( lookup863726980359885586_a_b_a @ St @ ( product_Pair_a_b @ Q4 @ Bs2 ) ) ) ) ) ).

% local.inv_def
thf(fact_78_prod__cases3,axiom,
    ! [Y5: produc1990351791824763820_d_nat] :
      ~ ! [A3: a,B3: c,C: option3893003681250311888_d_nat] :
          ( Y5
         != ( produc5865182755557626534_d_nat @ A3 @ ( produc5462365204240479225_d_nat @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_79_case__prodI2,axiom,
    ! [P2: produc3619439354540788382_b_a_a,C2: a > mappin4996136453398522101_b_a_a > $o] :
      ( ! [A3: a,B3: mappin4996136453398522101_b_a_a] :
          ( ( P2
            = ( produc4259011374121840526_b_a_a @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc5272303750545776437_a_a_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_80_case__prodI2,axiom,
    ! [P2: produc9176665399011442921_d_nat,C2: a > list_P6684558232516339260_d_nat > $o] :
      ( ! [A3: a,B3: list_P6684558232516339260_d_nat] :
          ( ( P2
            = ( produc919746265332163299_d_nat @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc968428876690328712_nat_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_81_case__prodI2,axiom,
    ! [P2: produc7083967959564351825_a_b_a,C2: a > list_P2992919015626629540_a_b_a > $o] :
      ( ! [A3: a,B3: list_P2992919015626629540_a_b_a] :
          ( ( P2
            = ( produc6994869176599099851_a_b_a @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc3087838301718062496_b_a_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_82_case__prodI2,axiom,
    ! [P2: produc2639161875728825943od_a_b,C2: a > list_P1467974523212632366od_a_b > $o] :
      ( ! [A3: a,B3: list_P1467974523212632366od_a_b] :
          ( ( P2
            = ( produc6643264353547226055od_a_b @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc7670699999246354108_a_b_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_83_case__prodI2,axiom,
    ! [P2: product_prod_a_a,C2: a > a > $o] :
      ( ! [A3: a,B3: a] :
          ( ( P2
            = ( product_Pair_a_a @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc6436628058953941356_a_a_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_84_case__prodI2,axiom,
    ! [P2: produc1990351791824763820_d_nat,C2: a > produc5456268773499131775_d_nat > $o] :
      ( ! [A3: a,B3: produc5456268773499131775_d_nat] :
          ( ( P2
            = ( produc5865182755557626534_d_nat @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc1905287380190322693_nat_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_85_case__prodI2,axiom,
    ! [P2: product_prod_a_b,C2: a > b > $o] :
      ( ! [A3: a,B3: b] :
          ( ( P2
            = ( product_Pair_a_b @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc3537405659489547051_a_b_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_86_case__prodI2,axiom,
    ! [P2: produc6343861673719254644_a_b_a,C2: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > $o] :
      ( ! [A3: produc1990351791824763820_d_nat,B3: mappin2365780136655028071_a_b_a] :
          ( ( P2
            = ( produc1681437058057156268_a_b_a @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc6407700812349938327_b_a_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_87_case__prodI2,axiom,
    ! [P2: produc2062766750192416900_a_b_a,C2: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > $o] :
      ( ! [A3: list_P6684558232516339260_d_nat,B3: mappin2365780136655028071_a_b_a] :
          ( ( P2
            = ( produc2616710729536263484_a_b_a @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc6164275942970966535_b_a_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_88_case__prodI2,axiom,
    ! [P2: produc7822699449245016212_a_b_a,C2: a > mappin2365780136655028071_a_b_a > $o] :
      ( ! [A3: a,B3: mappin2365780136655028071_a_b_a] :
          ( ( P2
            = ( produc1583556749610718734_a_b_a @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc2053338206788791453_b_a_o @ C2 @ P2 ) ) ).

% case_prodI2
thf(fact_89_case__prodI,axiom,
    ! [F: a > mappin4996136453398522101_b_a_a > $o,A: a,B: mappin4996136453398522101_b_a_a] :
      ( ( F @ A @ B )
     => ( produc5272303750545776437_a_a_o @ F @ ( produc4259011374121840526_b_a_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_90_case__prodI,axiom,
    ! [F: a > list_P6684558232516339260_d_nat > $o,A: a,B: list_P6684558232516339260_d_nat] :
      ( ( F @ A @ B )
     => ( produc968428876690328712_nat_o @ F @ ( produc919746265332163299_d_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_91_case__prodI,axiom,
    ! [F: a > list_P2992919015626629540_a_b_a > $o,A: a,B: list_P2992919015626629540_a_b_a] :
      ( ( F @ A @ B )
     => ( produc3087838301718062496_b_a_o @ F @ ( produc6994869176599099851_a_b_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_92_case__prodI,axiom,
    ! [F: a > list_P1467974523212632366od_a_b > $o,A: a,B: list_P1467974523212632366od_a_b] :
      ( ( F @ A @ B )
     => ( produc7670699999246354108_a_b_o @ F @ ( produc6643264353547226055od_a_b @ A @ B ) ) ) ).

% case_prodI
thf(fact_93_case__prodI,axiom,
    ! [F: a > a > $o,A: a,B: a] :
      ( ( F @ A @ B )
     => ( produc6436628058953941356_a_a_o @ F @ ( product_Pair_a_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_94_case__prodI,axiom,
    ! [F: a > produc5456268773499131775_d_nat > $o,A: a,B: produc5456268773499131775_d_nat] :
      ( ( F @ A @ B )
     => ( produc1905287380190322693_nat_o @ F @ ( produc5865182755557626534_d_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_95_case__prodI,axiom,
    ! [F: a > b > $o,A: a,B: b] :
      ( ( F @ A @ B )
     => ( produc3537405659489547051_a_b_o @ F @ ( product_Pair_a_b @ A @ B ) ) ) ).

% case_prodI
thf(fact_96_case__prodI,axiom,
    ! [F: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > $o,A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a] :
      ( ( F @ A @ B )
     => ( produc6407700812349938327_b_a_o @ F @ ( produc1681437058057156268_a_b_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_97_case__prodI,axiom,
    ! [F: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > $o,A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a] :
      ( ( F @ A @ B )
     => ( produc6164275942970966535_b_a_o @ F @ ( produc2616710729536263484_a_b_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_98_case__prodI,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > $o,A: a,B: mappin2365780136655028071_a_b_a] :
      ( ( F @ A @ B )
     => ( produc2053338206788791453_b_a_o @ F @ ( produc1583556749610718734_a_b_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_99_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_b,Z2: b,C2: a > b > set_b] :
      ( ! [A3: a,B3: b] :
          ( ( P2
            = ( product_Pair_a_b @ A3 @ B3 ) )
         => ( member_b @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_b @ Z2 @ ( produc1338550827010919922_set_b @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_100_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_b,Z2: a,C2: a > b > set_a] :
      ( ! [A3: a,B3: b] :
          ( ( P2
            = ( product_Pair_a_b @ A3 @ B3 ) )
         => ( member_a @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_a @ Z2 @ ( produc1338550822707691121_set_a @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_101_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_a,Z2: b,C2: a > a > set_b] :
      ( ! [A3: a,B3: a] :
          ( ( P2
            = ( product_Pair_a_a @ A3 @ B3 ) )
         => ( member_b @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_b @ Z2 @ ( produc9217457827056207795_set_b @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_102_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_a,Z2: a,C2: a > a > set_a] :
      ( ! [A3: a,B3: a] :
          ( ( P2
            = ( product_Pair_a_a @ A3 @ B3 ) )
         => ( member_a @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_a @ Z2 @ ( produc9217457822752978994_set_a @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_103_mem__case__prodI2,axiom,
    ! [P2: produc2639161875728825943od_a_b,Z2: b,C2: a > list_P1467974523212632366od_a_b > set_b] :
      ( ! [A3: a,B3: list_P1467974523212632366od_a_b] :
          ( ( P2
            = ( produc6643264353547226055od_a_b @ A3 @ B3 ) )
         => ( member_b @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_b @ Z2 @ ( produc1656899084517205251_set_b @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_104_mem__case__prodI2,axiom,
    ! [P2: produc2639161875728825943od_a_b,Z2: a,C2: a > list_P1467974523212632366od_a_b > set_a] :
      ( ! [A3: a,B3: list_P1467974523212632366od_a_b] :
          ( ( P2
            = ( produc6643264353547226055od_a_b @ A3 @ B3 ) )
         => ( member_a @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_a @ Z2 @ ( produc1656899080213976450_set_a @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_105_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_b,Z2: mappin2365780136655028071_a_b_a,C2: a > b > set_ma506406390364637853_a_b_a] :
      ( ! [A3: a,B3: b] :
          ( ( P2
            = ( product_Pair_a_b @ A3 @ B3 ) )
         => ( member2890181943093937918_a_b_a @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member2890181943093937918_a_b_a @ Z2 @ ( produc8201150708196116170_a_b_a @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_106_mem__case__prodI2,axiom,
    ! [P2: produc7822699449245016212_a_b_a,Z2: b,C2: a > mappin2365780136655028071_a_b_a > set_b] :
      ( ! [A3: a,B3: mappin2365780136655028071_a_b_a] :
          ( ( P2
            = ( produc1583556749610718734_a_b_a @ A3 @ B3 ) )
         => ( member_b @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_b @ Z2 @ ( produc367773994017072740_set_b @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_107_mem__case__prodI2,axiom,
    ! [P2: produc7822699449245016212_a_b_a,Z2: a,C2: a > mappin2365780136655028071_a_b_a > set_a] :
      ( ! [A3: a,B3: mappin2365780136655028071_a_b_a] :
          ( ( P2
            = ( produc1583556749610718734_a_b_a @ A3 @ B3 ) )
         => ( member_a @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_a @ Z2 @ ( produc367773989713843939_set_a @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_108_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_a,Z2: mappin2365780136655028071_a_b_a,C2: a > a > set_ma506406390364637853_a_b_a] :
      ( ! [A3: a,B3: a] :
          ( ( P2
            = ( product_Pair_a_a @ A3 @ B3 ) )
         => ( member2890181943093937918_a_b_a @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member2890181943093937918_a_b_a @ Z2 @ ( produc5751697835577802185_a_b_a @ C2 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_109_mem__case__prodI,axiom,
    ! [Z2: b,C2: a > b > set_b,A: a,B: b] :
      ( ( member_b @ Z2 @ ( C2 @ A @ B ) )
     => ( member_b @ Z2 @ ( produc1338550827010919922_set_b @ C2 @ ( product_Pair_a_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_110_mem__case__prodI,axiom,
    ! [Z2: a,C2: a > b > set_a,A: a,B: b] :
      ( ( member_a @ Z2 @ ( C2 @ A @ B ) )
     => ( member_a @ Z2 @ ( produc1338550822707691121_set_a @ C2 @ ( product_Pair_a_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_111_mem__case__prodI,axiom,
    ! [Z2: b,C2: a > a > set_b,A: a,B: a] :
      ( ( member_b @ Z2 @ ( C2 @ A @ B ) )
     => ( member_b @ Z2 @ ( produc9217457827056207795_set_b @ C2 @ ( product_Pair_a_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_112_mem__case__prodI,axiom,
    ! [Z2: a,C2: a > a > set_a,A: a,B: a] :
      ( ( member_a @ Z2 @ ( C2 @ A @ B ) )
     => ( member_a @ Z2 @ ( produc9217457822752978994_set_a @ C2 @ ( product_Pair_a_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_113_mem__case__prodI,axiom,
    ! [Z2: b,C2: a > list_P1467974523212632366od_a_b > set_b,A: a,B: list_P1467974523212632366od_a_b] :
      ( ( member_b @ Z2 @ ( C2 @ A @ B ) )
     => ( member_b @ Z2 @ ( produc1656899084517205251_set_b @ C2 @ ( produc6643264353547226055od_a_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_114_mem__case__prodI,axiom,
    ! [Z2: a,C2: a > list_P1467974523212632366od_a_b > set_a,A: a,B: list_P1467974523212632366od_a_b] :
      ( ( member_a @ Z2 @ ( C2 @ A @ B ) )
     => ( member_a @ Z2 @ ( produc1656899080213976450_set_a @ C2 @ ( produc6643264353547226055od_a_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_115_mem__case__prodI,axiom,
    ! [Z2: mappin2365780136655028071_a_b_a,C2: a > b > set_ma506406390364637853_a_b_a,A: a,B: b] :
      ( ( member2890181943093937918_a_b_a @ Z2 @ ( C2 @ A @ B ) )
     => ( member2890181943093937918_a_b_a @ Z2 @ ( produc8201150708196116170_a_b_a @ C2 @ ( product_Pair_a_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_116_mem__case__prodI,axiom,
    ! [Z2: b,C2: a > mappin2365780136655028071_a_b_a > set_b,A: a,B: mappin2365780136655028071_a_b_a] :
      ( ( member_b @ Z2 @ ( C2 @ A @ B ) )
     => ( member_b @ Z2 @ ( produc367773994017072740_set_b @ C2 @ ( produc1583556749610718734_a_b_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_117_mem__case__prodI,axiom,
    ! [Z2: a,C2: a > mappin2365780136655028071_a_b_a > set_a,A: a,B: mappin2365780136655028071_a_b_a] :
      ( ( member_a @ Z2 @ ( C2 @ A @ B ) )
     => ( member_a @ Z2 @ ( produc367773989713843939_set_a @ C2 @ ( produc1583556749610718734_a_b_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_118_mem__case__prodI,axiom,
    ! [Z2: mappin2365780136655028071_a_b_a,C2: a > a > set_ma506406390364637853_a_b_a,A: a,B: a] :
      ( ( member2890181943093937918_a_b_a @ Z2 @ ( C2 @ A @ B ) )
     => ( member2890181943093937918_a_b_a @ Z2 @ ( produc5751697835577802185_a_b_a @ C2 @ ( product_Pair_a_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_119_inv__st,axiom,
    inv @ st ).

% inv_st
thf(fact_120_fold_H,axiom,
    ( ( mmap_f6064999285056190608_a_b_a @ s @ st
      @ ( produc5613835436409813491_a_b_a
        @ ( produc5638855421663032305_a_b_a
          @ ^ [X4: a,V: produc5456268773499131775_d_nat,St: mappin2365780136655028071_a_b_a] :
              ( produc452744236373541113_a_b_a
              @ ^ [X5: a] : ( produc1681437058057156268_a_b_a @ ( produc5865182755557626534_d_nat @ X5 @ ( drop_cur_c_d @ i @ V ) ) )
              @ ( cstep_a_b @ step @ St @ X4 @ bs ) ) ) )
      @ ^ [X4: produc5456268773499131775_d_nat,Y4: produc5456268773499131775_d_nat] : X4
      @ r )
    = ( produc2616710729536263484_a_b_a @ s2 @ st2 ) ) ).

% fold'
thf(fact_121_mem__case__prodE,axiom,
    ! [Z2: b,C2: a > b > set_b,P2: product_prod_a_b] :
      ( ( member_b @ Z2 @ ( produc1338550827010919922_set_b @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: b] :
            ( ( P2
              = ( product_Pair_a_b @ X3 @ Y3 ) )
           => ~ ( member_b @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_122_mem__case__prodE,axiom,
    ! [Z2: a,C2: a > b > set_a,P2: product_prod_a_b] :
      ( ( member_a @ Z2 @ ( produc1338550822707691121_set_a @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: b] :
            ( ( P2
              = ( product_Pair_a_b @ X3 @ Y3 ) )
           => ~ ( member_a @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_123_mem__case__prodE,axiom,
    ! [Z2: b,C2: a > a > set_b,P2: product_prod_a_a] :
      ( ( member_b @ Z2 @ ( produc9217457827056207795_set_b @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X3 @ Y3 ) )
           => ~ ( member_b @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_124_mem__case__prodE,axiom,
    ! [Z2: a,C2: a > a > set_a,P2: product_prod_a_a] :
      ( ( member_a @ Z2 @ ( produc9217457822752978994_set_a @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X3 @ Y3 ) )
           => ~ ( member_a @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_125_mem__case__prodE,axiom,
    ! [Z2: b,C2: a > list_P1467974523212632366od_a_b > set_b,P2: produc2639161875728825943od_a_b] :
      ( ( member_b @ Z2 @ ( produc1656899084517205251_set_b @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: list_P1467974523212632366od_a_b] :
            ( ( P2
              = ( produc6643264353547226055od_a_b @ X3 @ Y3 ) )
           => ~ ( member_b @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_126_mem__case__prodE,axiom,
    ! [Z2: a,C2: a > list_P1467974523212632366od_a_b > set_a,P2: produc2639161875728825943od_a_b] :
      ( ( member_a @ Z2 @ ( produc1656899080213976450_set_a @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: list_P1467974523212632366od_a_b] :
            ( ( P2
              = ( produc6643264353547226055od_a_b @ X3 @ Y3 ) )
           => ~ ( member_a @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_127_mem__case__prodE,axiom,
    ! [Z2: mappin2365780136655028071_a_b_a,C2: a > b > set_ma506406390364637853_a_b_a,P2: product_prod_a_b] :
      ( ( member2890181943093937918_a_b_a @ Z2 @ ( produc8201150708196116170_a_b_a @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: b] :
            ( ( P2
              = ( product_Pair_a_b @ X3 @ Y3 ) )
           => ~ ( member2890181943093937918_a_b_a @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_128_mem__case__prodE,axiom,
    ! [Z2: b,C2: a > mappin2365780136655028071_a_b_a > set_b,P2: produc7822699449245016212_a_b_a] :
      ( ( member_b @ Z2 @ ( produc367773994017072740_set_b @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
            ( ( P2
              = ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) )
           => ~ ( member_b @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_129_mem__case__prodE,axiom,
    ! [Z2: a,C2: a > mappin2365780136655028071_a_b_a > set_a,P2: produc7822699449245016212_a_b_a] :
      ( ( member_a @ Z2 @ ( produc367773989713843939_set_a @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
            ( ( P2
              = ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) )
           => ~ ( member_a @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_130_mem__case__prodE,axiom,
    ! [Z2: mappin2365780136655028071_a_b_a,C2: a > a > set_ma506406390364637853_a_b_a,P2: product_prod_a_a] :
      ( ( member2890181943093937918_a_b_a @ Z2 @ ( produc5751697835577802185_a_b_a @ C2 @ P2 ) )
     => ~ ! [X3: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X3 @ Y3 ) )
           => ~ ( member2890181943093937918_a_b_a @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_131_case__prodE,axiom,
    ! [C2: a > mappin4996136453398522101_b_a_a > $o,P2: produc3619439354540788382_b_a_a] :
      ( ( produc5272303750545776437_a_a_o @ C2 @ P2 )
     => ~ ! [X3: a,Y3: mappin4996136453398522101_b_a_a] :
            ( ( P2
              = ( produc4259011374121840526_b_a_a @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_132_case__prodE,axiom,
    ! [C2: a > list_P6684558232516339260_d_nat > $o,P2: produc9176665399011442921_d_nat] :
      ( ( produc968428876690328712_nat_o @ C2 @ P2 )
     => ~ ! [X3: a,Y3: list_P6684558232516339260_d_nat] :
            ( ( P2
              = ( produc919746265332163299_d_nat @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_133_case__prodE,axiom,
    ! [C2: a > list_P2992919015626629540_a_b_a > $o,P2: produc7083967959564351825_a_b_a] :
      ( ( produc3087838301718062496_b_a_o @ C2 @ P2 )
     => ~ ! [X3: a,Y3: list_P2992919015626629540_a_b_a] :
            ( ( P2
              = ( produc6994869176599099851_a_b_a @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_134_case__prodE,axiom,
    ! [C2: a > list_P1467974523212632366od_a_b > $o,P2: produc2639161875728825943od_a_b] :
      ( ( produc7670699999246354108_a_b_o @ C2 @ P2 )
     => ~ ! [X3: a,Y3: list_P1467974523212632366od_a_b] :
            ( ( P2
              = ( produc6643264353547226055od_a_b @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_135_case__prodE,axiom,
    ! [C2: a > a > $o,P2: product_prod_a_a] :
      ( ( produc6436628058953941356_a_a_o @ C2 @ P2 )
     => ~ ! [X3: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_136_case__prodE,axiom,
    ! [C2: a > produc5456268773499131775_d_nat > $o,P2: produc1990351791824763820_d_nat] :
      ( ( produc1905287380190322693_nat_o @ C2 @ P2 )
     => ~ ! [X3: a,Y3: produc5456268773499131775_d_nat] :
            ( ( P2
              = ( produc5865182755557626534_d_nat @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_137_case__prodE,axiom,
    ! [C2: a > b > $o,P2: product_prod_a_b] :
      ( ( produc3537405659489547051_a_b_o @ C2 @ P2 )
     => ~ ! [X3: a,Y3: b] :
            ( ( P2
              = ( product_Pair_a_b @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_138_case__prodE,axiom,
    ! [C2: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > $o,P2: produc6343861673719254644_a_b_a] :
      ( ( produc6407700812349938327_b_a_o @ C2 @ P2 )
     => ~ ! [X3: produc1990351791824763820_d_nat,Y3: mappin2365780136655028071_a_b_a] :
            ( ( P2
              = ( produc1681437058057156268_a_b_a @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_139_case__prodE,axiom,
    ! [C2: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > $o,P2: produc2062766750192416900_a_b_a] :
      ( ( produc6164275942970966535_b_a_o @ C2 @ P2 )
     => ~ ! [X3: list_P6684558232516339260_d_nat,Y3: mappin2365780136655028071_a_b_a] :
            ( ( P2
              = ( produc2616710729536263484_a_b_a @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_140_case__prodE,axiom,
    ! [C2: a > mappin2365780136655028071_a_b_a > $o,P2: produc7822699449245016212_a_b_a] :
      ( ( produc2053338206788791453_b_a_o @ C2 @ P2 )
     => ~ ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
            ( ( P2
              = ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_141_case__prodD,axiom,
    ! [F: a > mappin4996136453398522101_b_a_a > $o,A: a,B: mappin4996136453398522101_b_a_a] :
      ( ( produc5272303750545776437_a_a_o @ F @ ( produc4259011374121840526_b_a_a @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_142_case__prodD,axiom,
    ! [F: a > list_P6684558232516339260_d_nat > $o,A: a,B: list_P6684558232516339260_d_nat] :
      ( ( produc968428876690328712_nat_o @ F @ ( produc919746265332163299_d_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_143_case__prodD,axiom,
    ! [F: a > list_P2992919015626629540_a_b_a > $o,A: a,B: list_P2992919015626629540_a_b_a] :
      ( ( produc3087838301718062496_b_a_o @ F @ ( produc6994869176599099851_a_b_a @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_144_case__prodD,axiom,
    ! [F: a > list_P1467974523212632366od_a_b > $o,A: a,B: list_P1467974523212632366od_a_b] :
      ( ( produc7670699999246354108_a_b_o @ F @ ( produc6643264353547226055od_a_b @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_145_case__prodD,axiom,
    ! [F: a > a > $o,A: a,B: a] :
      ( ( produc6436628058953941356_a_a_o @ F @ ( product_Pair_a_a @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_146_case__prodD,axiom,
    ! [F: a > produc5456268773499131775_d_nat > $o,A: a,B: produc5456268773499131775_d_nat] :
      ( ( produc1905287380190322693_nat_o @ F @ ( produc5865182755557626534_d_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_147_case__prodD,axiom,
    ! [F: a > b > $o,A: a,B: b] :
      ( ( produc3537405659489547051_a_b_o @ F @ ( product_Pair_a_b @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_148_case__prodD,axiom,
    ! [F: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > $o,A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a] :
      ( ( produc6407700812349938327_b_a_o @ F @ ( produc1681437058057156268_a_b_a @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_149_case__prodD,axiom,
    ! [F: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > $o,A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a] :
      ( ( produc6164275942970966535_b_a_o @ F @ ( produc2616710729536263484_a_b_a @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_150_case__prodD,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > $o,A: a,B: mappin2365780136655028071_a_b_a] :
      ( ( produc2053338206788791453_b_a_o @ F @ ( produc1583556749610718734_a_b_a @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_151_Pair__inject,axiom,
    ! [A: a,B: mappin4996136453398522101_b_a_a,A2: a,B2: mappin4996136453398522101_b_a_a] :
      ( ( ( produc4259011374121840526_b_a_a @ A @ B )
        = ( produc4259011374121840526_b_a_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_152_Pair__inject,axiom,
    ! [A: a,B: list_P6684558232516339260_d_nat,A2: a,B2: list_P6684558232516339260_d_nat] :
      ( ( ( produc919746265332163299_d_nat @ A @ B )
        = ( produc919746265332163299_d_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_153_Pair__inject,axiom,
    ! [A: a,B: list_P2992919015626629540_a_b_a,A2: a,B2: list_P2992919015626629540_a_b_a] :
      ( ( ( produc6994869176599099851_a_b_a @ A @ B )
        = ( produc6994869176599099851_a_b_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_154_Pair__inject,axiom,
    ! [A: a,B: list_P1467974523212632366od_a_b,A2: a,B2: list_P1467974523212632366od_a_b] :
      ( ( ( produc6643264353547226055od_a_b @ A @ B )
        = ( produc6643264353547226055od_a_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_155_Pair__inject,axiom,
    ! [A: a,B: a,A2: a,B2: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_156_Pair__inject,axiom,
    ! [A: a,B: produc5456268773499131775_d_nat,A2: a,B2: produc5456268773499131775_d_nat] :
      ( ( ( produc5865182755557626534_d_nat @ A @ B )
        = ( produc5865182755557626534_d_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_157_Pair__inject,axiom,
    ! [A: a,B: b,A2: a,B2: b] :
      ( ( ( product_Pair_a_b @ A @ B )
        = ( product_Pair_a_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_158_Pair__inject,axiom,
    ! [A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a,A2: produc1990351791824763820_d_nat,B2: mappin2365780136655028071_a_b_a] :
      ( ( ( produc1681437058057156268_a_b_a @ A @ B )
        = ( produc1681437058057156268_a_b_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_159_Pair__inject,axiom,
    ! [A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a,A2: list_P6684558232516339260_d_nat,B2: mappin2365780136655028071_a_b_a] :
      ( ( ( produc2616710729536263484_a_b_a @ A @ B )
        = ( produc2616710729536263484_a_b_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_160_Pair__inject,axiom,
    ! [A: a,B: mappin2365780136655028071_a_b_a,A2: a,B2: mappin2365780136655028071_a_b_a] :
      ( ( ( produc1583556749610718734_a_b_a @ A @ B )
        = ( produc1583556749610718734_a_b_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_161_prod__cases,axiom,
    ! [P: produc3619439354540788382_b_a_a > $o,P2: produc3619439354540788382_b_a_a] :
      ( ! [A3: a,B3: mappin4996136453398522101_b_a_a] : ( P @ ( produc4259011374121840526_b_a_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_162_prod__cases,axiom,
    ! [P: produc9176665399011442921_d_nat > $o,P2: produc9176665399011442921_d_nat] :
      ( ! [A3: a,B3: list_P6684558232516339260_d_nat] : ( P @ ( produc919746265332163299_d_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_163_prod__cases,axiom,
    ! [P: produc7083967959564351825_a_b_a > $o,P2: produc7083967959564351825_a_b_a] :
      ( ! [A3: a,B3: list_P2992919015626629540_a_b_a] : ( P @ ( produc6994869176599099851_a_b_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_164_prod__cases,axiom,
    ! [P: produc2639161875728825943od_a_b > $o,P2: produc2639161875728825943od_a_b] :
      ( ! [A3: a,B3: list_P1467974523212632366od_a_b] : ( P @ ( produc6643264353547226055od_a_b @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_165_prod__cases,axiom,
    ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
      ( ! [A3: a,B3: a] : ( P @ ( product_Pair_a_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_166_prod__cases,axiom,
    ! [P: produc1990351791824763820_d_nat > $o,P2: produc1990351791824763820_d_nat] :
      ( ! [A3: a,B3: produc5456268773499131775_d_nat] : ( P @ ( produc5865182755557626534_d_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_167_prod__cases,axiom,
    ! [P: product_prod_a_b > $o,P2: product_prod_a_b] :
      ( ! [A3: a,B3: b] : ( P @ ( product_Pair_a_b @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_168_prod__cases,axiom,
    ! [P: produc6343861673719254644_a_b_a > $o,P2: produc6343861673719254644_a_b_a] :
      ( ! [A3: produc1990351791824763820_d_nat,B3: mappin2365780136655028071_a_b_a] : ( P @ ( produc1681437058057156268_a_b_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_169_prod__cases,axiom,
    ! [P: produc2062766750192416900_a_b_a > $o,P2: produc2062766750192416900_a_b_a] :
      ( ! [A3: list_P6684558232516339260_d_nat,B3: mappin2365780136655028071_a_b_a] : ( P @ ( produc2616710729536263484_a_b_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_170_prod__cases,axiom,
    ! [P: produc7822699449245016212_a_b_a > $o,P2: produc7822699449245016212_a_b_a] :
      ( ! [A3: a,B3: mappin2365780136655028071_a_b_a] : ( P @ ( produc1583556749610718734_a_b_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_171_surj__pair,axiom,
    ! [P2: produc3619439354540788382_b_a_a] :
    ? [X3: a,Y3: mappin4996136453398522101_b_a_a] :
      ( P2
      = ( produc4259011374121840526_b_a_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_172_surj__pair,axiom,
    ! [P2: produc9176665399011442921_d_nat] :
    ? [X3: a,Y3: list_P6684558232516339260_d_nat] :
      ( P2
      = ( produc919746265332163299_d_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_173_surj__pair,axiom,
    ! [P2: produc7083967959564351825_a_b_a] :
    ? [X3: a,Y3: list_P2992919015626629540_a_b_a] :
      ( P2
      = ( produc6994869176599099851_a_b_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_174_surj__pair,axiom,
    ! [P2: produc2639161875728825943od_a_b] :
    ? [X3: a,Y3: list_P1467974523212632366od_a_b] :
      ( P2
      = ( produc6643264353547226055od_a_b @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_175_surj__pair,axiom,
    ! [P2: product_prod_a_a] :
    ? [X3: a,Y3: a] :
      ( P2
      = ( product_Pair_a_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_176_surj__pair,axiom,
    ! [P2: produc1990351791824763820_d_nat] :
    ? [X3: a,Y3: produc5456268773499131775_d_nat] :
      ( P2
      = ( produc5865182755557626534_d_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_177_surj__pair,axiom,
    ! [P2: product_prod_a_b] :
    ? [X3: a,Y3: b] :
      ( P2
      = ( product_Pair_a_b @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_178_surj__pair,axiom,
    ! [P2: produc6343861673719254644_a_b_a] :
    ? [X3: produc1990351791824763820_d_nat,Y3: mappin2365780136655028071_a_b_a] :
      ( P2
      = ( produc1681437058057156268_a_b_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_179_surj__pair,axiom,
    ! [P2: produc2062766750192416900_a_b_a] :
    ? [X3: list_P6684558232516339260_d_nat,Y3: mappin2365780136655028071_a_b_a] :
      ( P2
      = ( produc2616710729536263484_a_b_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_180_surj__pair,axiom,
    ! [P2: produc7822699449245016212_a_b_a] :
    ? [X3: a,Y3: mappin2365780136655028071_a_b_a] :
      ( P2
      = ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_181_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc3619439354540788382_b_a_a] :
      ~ ! [A3: a,B3: mappin4996136453398522101_b_a_a] :
          ( Y5
         != ( produc4259011374121840526_b_a_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_182_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc9176665399011442921_d_nat] :
      ~ ! [A3: a,B3: list_P6684558232516339260_d_nat] :
          ( Y5
         != ( produc919746265332163299_d_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_183_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc7083967959564351825_a_b_a] :
      ~ ! [A3: a,B3: list_P2992919015626629540_a_b_a] :
          ( Y5
         != ( produc6994869176599099851_a_b_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_184_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc2639161875728825943od_a_b] :
      ~ ! [A3: a,B3: list_P1467974523212632366od_a_b] :
          ( Y5
         != ( produc6643264353547226055od_a_b @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_185_old_Oprod_Oexhaust,axiom,
    ! [Y5: product_prod_a_a] :
      ~ ! [A3: a,B3: a] :
          ( Y5
         != ( product_Pair_a_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_186_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc1990351791824763820_d_nat] :
      ~ ! [A3: a,B3: produc5456268773499131775_d_nat] :
          ( Y5
         != ( produc5865182755557626534_d_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_187_old_Oprod_Oexhaust,axiom,
    ! [Y5: product_prod_a_b] :
      ~ ! [A3: a,B3: b] :
          ( Y5
         != ( product_Pair_a_b @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_188_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc6343861673719254644_a_b_a] :
      ~ ! [A3: produc1990351791824763820_d_nat,B3: mappin2365780136655028071_a_b_a] :
          ( Y5
         != ( produc1681437058057156268_a_b_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_189_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc2062766750192416900_a_b_a] :
      ~ ! [A3: list_P6684558232516339260_d_nat,B3: mappin2365780136655028071_a_b_a] :
          ( Y5
         != ( produc2616710729536263484_a_b_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_190_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc7822699449245016212_a_b_a] :
      ~ ! [A3: a,B3: mappin2365780136655028071_a_b_a] :
          ( Y5
         != ( produc1583556749610718734_a_b_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_191_case__prod__app,axiom,
    ( produc5638855421663032305_a_b_a
    = ( ^ [F2: a > produc5456268773499131775_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] :
          ( produc8695831730334998625_a_b_a
          @ ^ [L: a,R2: produc5456268773499131775_d_nat] : ( F2 @ L @ R2 @ Y4 )
          @ X4 ) ) ) ).

% case_prod_app
thf(fact_192_prod_Ocase__distrib,axiom,
    ! [H: product_prod_a_b > product_prod_a_b,F: a > b > product_prod_a_b,Prod: product_prod_a_b] :
      ( ( H @ ( produc8992199381948149691od_a_b @ F @ Prod ) )
      = ( produc8992199381948149691od_a_b
        @ ^ [X12: a,X22: b] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_193_prod_Ocase__distrib,axiom,
    ! [H: produc1990351791824763820_d_nat > produc1990351791824763820_d_nat,F: a > produc5456268773499131775_d_nat > produc1990351791824763820_d_nat,Prod: produc1990351791824763820_d_nat] :
      ( ( H @ ( produc5272039204407661311_d_nat @ F @ Prod ) )
      = ( produc5272039204407661311_d_nat
        @ ^ [X12: a,X22: produc5456268773499131775_d_nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_194_prod_Ocase__distrib,axiom,
    ! [H: produc6343861673719254644_a_b_a > produc6343861673719254644_a_b_a,F: a > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,Prod: produc7822699449245016212_a_b_a] :
      ( ( H @ ( produc452744236373541113_a_b_a @ F @ Prod ) )
      = ( produc452744236373541113_a_b_a
        @ ^ [X12: a,X22: mappin2365780136655028071_a_b_a] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_195_prod_Ocase__distrib,axiom,
    ! [H: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,F: a > produc5456268773499131775_d_nat > produc1990351791824763820_d_nat,Prod: produc1990351791824763820_d_nat] :
      ( ( H @ ( produc5272039204407661311_d_nat @ F @ Prod ) )
      = ( produc5638855421663032305_a_b_a
        @ ^ [X12: a,X22: produc5456268773499131775_d_nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_196_prod_Ocase__distrib,axiom,
    ! [H: ( mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a ) > produc1990351791824763820_d_nat,F: a > produc5456268773499131775_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,Prod: produc1990351791824763820_d_nat] :
      ( ( H @ ( produc5638855421663032305_a_b_a @ F @ Prod ) )
      = ( produc5272039204407661311_d_nat
        @ ^ [X12: a,X22: produc5456268773499131775_d_nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_197_prod_Ocase__distrib,axiom,
    ! [H: produc6343861673719254644_a_b_a > produc6343861673719254644_a_b_a,F: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,Prod: produc6343861673719254644_a_b_a] :
      ( ( H @ ( produc5613835436409813491_a_b_a @ F @ Prod ) )
      = ( produc5613835436409813491_a_b_a
        @ ^ [X12: produc1990351791824763820_d_nat,X22: mappin2365780136655028071_a_b_a] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_198_prod_Ocase__distrib,axiom,
    ! [H: produc2062766750192416900_a_b_a > produc2062766750192416900_a_b_a,F: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > produc2062766750192416900_a_b_a,Prod: produc2062766750192416900_a_b_a] :
      ( ( H @ ( produc3256888519411912691_a_b_a @ F @ Prod ) )
      = ( produc3256888519411912691_a_b_a
        @ ^ [X12: list_P6684558232516339260_d_nat,X22: mappin2365780136655028071_a_b_a] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_199_prod_Ocase__distrib,axiom,
    ! [H: ( mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a ) > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,F: a > produc5456268773499131775_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,Prod: produc1990351791824763820_d_nat] :
      ( ( H @ ( produc5638855421663032305_a_b_a @ F @ Prod ) )
      = ( produc5638855421663032305_a_b_a
        @ ^ [X12: a,X22: produc5456268773499131775_d_nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_200_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: a > b > $o,Prod: product_prod_a_b] :
      ( ( H @ ( produc3537405659489547051_a_b_o @ F @ Prod ) )
      = ( produc3537405659489547051_a_b_o
        @ ^ [X12: a,X22: b] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_201_prod_Ocase__distrib,axiom,
    ! [H: product_prod_a_b > $o,F: a > b > product_prod_a_b,Prod: product_prod_a_b] :
      ( ( H @ ( produc8992199381948149691od_a_b @ F @ Prod ) )
      = ( produc3537405659489547051_a_b_o
        @ ^ [X12: a,X22: b] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_202_old_Oprod_Ocase,axiom,
    ! [F: a > b > product_prod_a_b,X1: a,X2: b] :
      ( ( produc8992199381948149691od_a_b @ F @ ( product_Pair_a_b @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_203_old_Oprod_Ocase,axiom,
    ! [F: a > produc5456268773499131775_d_nat > produc1990351791824763820_d_nat,X1: a,X2: produc5456268773499131775_d_nat] :
      ( ( produc5272039204407661311_d_nat @ F @ ( produc5865182755557626534_d_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_204_old_Oprod_Ocase,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,X1: a,X2: mappin2365780136655028071_a_b_a] :
      ( ( produc452744236373541113_a_b_a @ F @ ( produc1583556749610718734_a_b_a @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_205_old_Oprod_Ocase,axiom,
    ! [F: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,X1: produc1990351791824763820_d_nat,X2: mappin2365780136655028071_a_b_a] :
      ( ( produc5613835436409813491_a_b_a @ F @ ( produc1681437058057156268_a_b_a @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_206_old_Oprod_Ocase,axiom,
    ! [F: a > produc5456268773499131775_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,X1: a,X2: produc5456268773499131775_d_nat] :
      ( ( produc5638855421663032305_a_b_a @ F @ ( produc5865182755557626534_d_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_207_old_Oprod_Ocase,axiom,
    ! [F: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > produc2062766750192416900_a_b_a,X1: list_P6684558232516339260_d_nat,X2: mappin2365780136655028071_a_b_a] :
      ( ( produc3256888519411912691_a_b_a @ F @ ( produc2616710729536263484_a_b_a @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_208_old_Oprod_Ocase,axiom,
    ! [F: a > b > $o,X1: a,X2: b] :
      ( ( produc3537405659489547051_a_b_o @ F @ ( product_Pair_a_b @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_209_old_Oprod_Ocase,axiom,
    ! [F: a > b > list_P1467974523212632366od_a_b,X1: a,X2: b] :
      ( ( produc611504906096598721od_a_b @ F @ ( product_Pair_a_b @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_210_old_Oprod_Ocase,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > $o,X1: a,X2: mappin2365780136655028071_a_b_a] :
      ( ( produc2053338206788791453_b_a_o @ F @ ( produc1583556749610718734_a_b_a @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_211_old_Oprod_Ocase,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > produc7822699449245016212_a_b_a,X1: a,X2: mappin2365780136655028071_a_b_a] :
      ( ( produc543687385667932879_a_b_a @ F @ ( produc1583556749610718734_a_b_a @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_212_prod__induct3,axiom,
    ! [P: produc1990351791824763820_d_nat > $o,X: produc1990351791824763820_d_nat] :
      ( ! [A3: a,B3: c,C: option3893003681250311888_d_nat] : ( P @ ( produc5865182755557626534_d_nat @ A3 @ ( produc5462365204240479225_d_nat @ B3 @ C ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_213_mmap__fold_H__eq,axiom,
    ! [M: list_P6684558232516339260_d_nat,E: mappin2365780136655028071_a_b_a,F3: produc6343861673719254644_a_b_a > produc6343861673719254644_a_b_a,C2: produc5456268773499131775_d_nat > produc5456268773499131775_d_nat > produc5456268773499131775_d_nat,R3: list_P6684558232516339260_d_nat,M2: list_P6684558232516339260_d_nat,E2: mappin2365780136655028071_a_b_a,P: mappin2365780136655028071_a_b_a > $o,F: produc1990351791824763820_d_nat > produc1990351791824763820_d_nat] :
      ( ( ( mmap_f6064999285056190608_a_b_a @ M @ E @ F3 @ C2 @ R3 )
        = ( produc2616710729536263484_a_b_a @ M2 @ E2 ) )
     => ( ( P @ E )
       => ( ! [P3: produc1990351791824763820_d_nat,E3: mappin2365780136655028071_a_b_a,P4: produc1990351791824763820_d_nat,E4: mappin2365780136655028071_a_b_a] :
              ( ( P @ E3 )
             => ( ( ( F3 @ ( produc1681437058057156268_a_b_a @ P3 @ E3 ) )
                  = ( produc1681437058057156268_a_b_a @ P4 @ E4 ) )
               => ( ( P4
                    = ( F @ P3 ) )
                  & ( P @ E4 ) ) ) )
         => ( ( M2
              = ( mmap_f8625263917883502961_d_nat @ M @ F @ C2 @ R3 ) )
            & ( P @ E2 ) ) ) ) ) ).

% mmap_fold'_eq
thf(fact_214_mem__Collect__eq,axiom,
    ! [A: list_P6684558232516339260_d_nat,P: list_P6684558232516339260_d_nat > $o] :
      ( ( member136131075555046355_d_nat @ A @ ( collec3725658159922174225_d_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_215_mem__Collect__eq,axiom,
    ! [A: mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,P: ( mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a ) > $o] :
      ( ( member4644490972414488365_a_b_a @ A @ ( collec8070199481768543855_a_b_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_216_mem__Collect__eq,axiom,
    ! [A: a,P: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_217_mem__Collect__eq,axiom,
    ! [A: produc1990351791824763820_d_nat,P: produc1990351791824763820_d_nat > $o] :
      ( ( member6046481508403577027_d_nat @ A @ ( collec3895662986305461505_d_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_218_mem__Collect__eq,axiom,
    ! [A: produc7822699449245016212_a_b_a,P: produc7822699449245016212_a_b_a > $o] :
      ( ( member3694737009695996203_a_b_a @ A @ ( collec8037893338069451881_a_b_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_219_mem__Collect__eq,axiom,
    ! [A: product_prod_a_b,P: product_prod_a_b > $o] :
      ( ( member1426531481828664017od_a_b @ A @ ( collec3336397801687681299od_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_220_mem__Collect__eq,axiom,
    ! [A: mappin2365780136655028071_a_b_a,P: mappin2365780136655028071_a_b_a > $o] :
      ( ( member2890181943093937918_a_b_a @ A @ ( collec6734253767060832316_a_b_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_221_mem__Collect__eq,axiom,
    ! [A: b,P: b > $o] :
      ( ( member_b @ A @ ( collect_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_222_Collect__mem__eq,axiom,
    ! [A4: set_li2034412094485292274_d_nat] :
      ( ( collec3725658159922174225_d_nat
        @ ^ [X4: list_P6684558232516339260_d_nat] : ( member136131075555046355_d_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_223_Collect__mem__eq,axiom,
    ! [A4: set_ma3755819940569790564_a_b_a] :
      ( ( collec8070199481768543855_a_b_a
        @ ^ [X4: mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a] : ( member4644490972414488365_a_b_a @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_224_Collect__mem__eq,axiom,
    ! [A4: set_a] :
      ( ( collect_a
        @ ^ [X4: a] : ( member_a @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_225_Collect__mem__eq,axiom,
    ! [A4: set_Pr8583615357880026082_d_nat] :
      ( ( collec3895662986305461505_d_nat
        @ ^ [X4: produc1990351791824763820_d_nat] : ( member6046481508403577027_d_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_226_Collect__mem__eq,axiom,
    ! [A4: set_Pr2084289749507318346_a_b_a] :
      ( ( collec8037893338069451881_a_b_a
        @ ^ [X4: produc7822699449245016212_a_b_a] : ( member3694737009695996203_a_b_a @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_227_Collect__mem__eq,axiom,
    ! [A4: set_Product_prod_a_b] :
      ( ( collec3336397801687681299od_a_b
        @ ^ [X4: product_prod_a_b] : ( member1426531481828664017od_a_b @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_228_Collect__mem__eq,axiom,
    ! [A4: set_ma506406390364637853_a_b_a] :
      ( ( collec6734253767060832316_a_b_a
        @ ^ [X4: mappin2365780136655028071_a_b_a] : ( member2890181943093937918_a_b_a @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_229_Collect__mem__eq,axiom,
    ! [A4: set_b] :
      ( ( collect_b
        @ ^ [X4: b] : ( member_b @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_230_Collect__cong,axiom,
    ! [P: produc1990351791824763820_d_nat > $o,Q2: produc1990351791824763820_d_nat > $o] :
      ( ! [X3: produc1990351791824763820_d_nat] :
          ( ( P @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collec3895662986305461505_d_nat @ P )
        = ( collec3895662986305461505_d_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_231_Collect__cong,axiom,
    ! [P: produc7822699449245016212_a_b_a > $o,Q2: produc7822699449245016212_a_b_a > $o] :
      ( ! [X3: produc7822699449245016212_a_b_a] :
          ( ( P @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collec8037893338069451881_a_b_a @ P )
        = ( collec8037893338069451881_a_b_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_232_Collect__cong,axiom,
    ! [P: product_prod_a_b > $o,Q2: product_prod_a_b > $o] :
      ( ! [X3: product_prod_a_b] :
          ( ( P @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collec3336397801687681299od_a_b @ P )
        = ( collec3336397801687681299od_a_b @ Q2 ) ) ) ).

% Collect_cong
thf(fact_233_Collect__cong,axiom,
    ! [P: mappin2365780136655028071_a_b_a > $o,Q2: mappin2365780136655028071_a_b_a > $o] :
      ( ! [X3: mappin2365780136655028071_a_b_a] :
          ( ( P @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collec6734253767060832316_a_b_a @ P )
        = ( collec6734253767060832316_a_b_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_234_Collect__cong,axiom,
    ! [P: b > $o,Q2: b > $o] :
      ( ! [X3: b] :
          ( ( P @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_b @ P )
        = ( collect_b @ Q2 ) ) ) ).

% Collect_cong
thf(fact_235_split__cong,axiom,
    ! [Q: product_prod_a_b,F: a > b > product_prod_a_b,G: a > b > product_prod_a_b,P2: product_prod_a_b] :
      ( ! [X3: a,Y3: b] :
          ( ( ( product_Pair_a_b @ X3 @ Y3 )
            = Q )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q )
       => ( ( produc8992199381948149691od_a_b @ F @ P2 )
          = ( produc8992199381948149691od_a_b @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_236_split__cong,axiom,
    ! [Q: produc1990351791824763820_d_nat,F: a > produc5456268773499131775_d_nat > produc1990351791824763820_d_nat,G: a > produc5456268773499131775_d_nat > produc1990351791824763820_d_nat,P2: produc1990351791824763820_d_nat] :
      ( ! [X3: a,Y3: produc5456268773499131775_d_nat] :
          ( ( ( produc5865182755557626534_d_nat @ X3 @ Y3 )
            = Q )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q )
       => ( ( produc5272039204407661311_d_nat @ F @ P2 )
          = ( produc5272039204407661311_d_nat @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_237_split__cong,axiom,
    ! [Q: produc7822699449245016212_a_b_a,F: a > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,G: a > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,P2: produc7822699449245016212_a_b_a] :
      ( ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
          ( ( ( produc1583556749610718734_a_b_a @ X3 @ Y3 )
            = Q )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q )
       => ( ( produc452744236373541113_a_b_a @ F @ P2 )
          = ( produc452744236373541113_a_b_a @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_238_split__cong,axiom,
    ! [Q: produc6343861673719254644_a_b_a,F: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,G: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,P2: produc6343861673719254644_a_b_a] :
      ( ! [X3: produc1990351791824763820_d_nat,Y3: mappin2365780136655028071_a_b_a] :
          ( ( ( produc1681437058057156268_a_b_a @ X3 @ Y3 )
            = Q )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q )
       => ( ( produc5613835436409813491_a_b_a @ F @ P2 )
          = ( produc5613835436409813491_a_b_a @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_239_split__cong,axiom,
    ! [Q: produc1990351791824763820_d_nat,F: a > produc5456268773499131775_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,G: a > produc5456268773499131775_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,P2: produc1990351791824763820_d_nat] :
      ( ! [X3: a,Y3: produc5456268773499131775_d_nat] :
          ( ( ( produc5865182755557626534_d_nat @ X3 @ Y3 )
            = Q )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q )
       => ( ( produc5638855421663032305_a_b_a @ F @ P2 )
          = ( produc5638855421663032305_a_b_a @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_240_split__cong,axiom,
    ! [Q: produc2062766750192416900_a_b_a,F: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > produc2062766750192416900_a_b_a,G: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > produc2062766750192416900_a_b_a,P2: produc2062766750192416900_a_b_a] :
      ( ! [X3: list_P6684558232516339260_d_nat,Y3: mappin2365780136655028071_a_b_a] :
          ( ( ( produc2616710729536263484_a_b_a @ X3 @ Y3 )
            = Q )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q )
       => ( ( produc3256888519411912691_a_b_a @ F @ P2 )
          = ( produc3256888519411912691_a_b_a @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_241_split__cong,axiom,
    ! [Q: product_prod_a_b,F: a > b > $o,G: a > b > $o,P2: product_prod_a_b] :
      ( ! [X3: a,Y3: b] :
          ( ( ( product_Pair_a_b @ X3 @ Y3 )
            = Q )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q )
       => ( ( produc3537405659489547051_a_b_o @ F @ P2 )
          = ( produc3537405659489547051_a_b_o @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_242_split__cong,axiom,
    ! [Q: product_prod_a_b,F: a > b > list_P1467974523212632366od_a_b,G: a > b > list_P1467974523212632366od_a_b,P2: product_prod_a_b] :
      ( ! [X3: a,Y3: b] :
          ( ( ( product_Pair_a_b @ X3 @ Y3 )
            = Q )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q )
       => ( ( produc611504906096598721od_a_b @ F @ P2 )
          = ( produc611504906096598721od_a_b @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_243_split__cong,axiom,
    ! [Q: produc7822699449245016212_a_b_a,F: a > mappin2365780136655028071_a_b_a > $o,G: a > mappin2365780136655028071_a_b_a > $o,P2: produc7822699449245016212_a_b_a] :
      ( ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
          ( ( ( produc1583556749610718734_a_b_a @ X3 @ Y3 )
            = Q )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q )
       => ( ( produc2053338206788791453_b_a_o @ F @ P2 )
          = ( produc2053338206788791453_b_a_o @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_244_split__cong,axiom,
    ! [Q: produc7822699449245016212_a_b_a,F: a > mappin2365780136655028071_a_b_a > produc7822699449245016212_a_b_a,G: a > mappin2365780136655028071_a_b_a > produc7822699449245016212_a_b_a,P2: produc7822699449245016212_a_b_a] :
      ( ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
          ( ( ( produc1583556749610718734_a_b_a @ X3 @ Y3 )
            = Q )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q )
       => ( ( produc543687385667932879_a_b_a @ F @ P2 )
          = ( produc543687385667932879_a_b_a @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_245_option_Ocase__distrib,axiom,
    ! [H: $o > $o,F1: $o,F22: a > $o,Option: option_a] :
      ( ( H @ ( case_option_o_a @ F1 @ F22 @ Option ) )
      = ( case_option_o_a @ ( H @ F1 )
        @ ^ [X4: a] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_246_option_Ocase__distrib,axiom,
    ! [H: $o > a,F1: $o,F22: a > $o,Option: option_a] :
      ( ( H @ ( case_option_o_a @ F1 @ F22 @ Option ) )
      = ( case_option_a_a @ ( H @ F1 )
        @ ^ [X4: a] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_247_option_Ocase__distrib,axiom,
    ! [H: a > $o,F1: a,F22: a > a,Option: option_a] :
      ( ( H @ ( case_option_a_a @ F1 @ F22 @ Option ) )
      = ( case_option_o_a @ ( H @ F1 )
        @ ^ [X4: a] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_248_option_Ocase__distrib,axiom,
    ! [H: a > a,F1: a,F22: a > a,Option: option_a] :
      ( ( H @ ( case_option_a_a @ F1 @ F22 @ Option ) )
      = ( case_option_a_a @ ( H @ F1 )
        @ ^ [X4: a] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_249_option_Ocase__distrib,axiom,
    ! [H: $o > option_a,F1: $o,F22: a > $o,Option: option_a] :
      ( ( H @ ( case_option_o_a @ F1 @ F22 @ Option ) )
      = ( case_o3148979394504432965on_a_a @ ( H @ F1 )
        @ ^ [X4: a] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_250_option_Ocase__distrib,axiom,
    ! [H: option_a > $o,F1: option_a,F22: a > option_a,Option: option_a] :
      ( ( H @ ( case_o3148979394504432965on_a_a @ F1 @ F22 @ Option ) )
      = ( case_option_o_a @ ( H @ F1 )
        @ ^ [X4: a] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_251_option_Ocase__distrib,axiom,
    ! [H: option_a > a,F1: option_a,F22: a > option_a,Option: option_a] :
      ( ( H @ ( case_o3148979394504432965on_a_a @ F1 @ F22 @ Option ) )
      = ( case_option_a_a @ ( H @ F1 )
        @ ^ [X4: a] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_252_option_Ocase__distrib,axiom,
    ! [H: a > option_a,F1: a,F22: a > a,Option: option_a] :
      ( ( H @ ( case_option_a_a @ F1 @ F22 @ Option ) )
      = ( case_o3148979394504432965on_a_a @ ( H @ F1 )
        @ ^ [X4: a] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_253_option_Ocase__distrib,axiom,
    ! [H: option_a > option_a,F1: option_a,F22: a > option_a,Option: option_a] :
      ( ( H @ ( case_o3148979394504432965on_a_a @ F1 @ F22 @ Option ) )
      = ( case_o3148979394504432965on_a_a @ ( H @ F1 )
        @ ^ [X4: a] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_254_option_Ocase__distrib,axiom,
    ! [H: $o > mappin2365780136655028071_a_b_a,F1: $o,F22: a > $o,Option: option_a] :
      ( ( H @ ( case_option_o_a @ F1 @ F22 @ Option ) )
      = ( case_o2154183385986904842_b_a_a @ ( H @ F1 )
        @ ^ [X4: a] : ( H @ ( F22 @ X4 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_255_internal__case__prod__def,axiom,
    produc3797153600616317541od_a_b = produc8992199381948149691od_a_b ).

% internal_case_prod_def
thf(fact_256_internal__case__prod__def,axiom,
    produc3580122457448217769_d_nat = produc5272039204407661311_d_nat ).

% internal_case_prod_def
thf(fact_257_internal__case__prod__def,axiom,
    produc5628374565148576719_a_b_a = produc452744236373541113_a_b_a ).

% internal_case_prod_def
thf(fact_258_internal__case__prod__def,axiom,
    produc4382883301709631645_a_b_a = produc5613835436409813491_a_b_a ).

% internal_case_prod_def
thf(fact_259_internal__case__prod__def,axiom,
    produc4871049226530554055_a_b_a = produc5638855421663032305_a_b_a ).

% internal_case_prod_def
thf(fact_260_internal__case__prod__def,axiom,
    produc8391920324087245981_a_b_a = produc3256888519411912691_a_b_a ).

% internal_case_prod_def
thf(fact_261_internal__case__prod__def,axiom,
    produc4617426408912073941_a_b_o = produc3537405659489547051_a_b_o ).

% internal_case_prod_def
thf(fact_262_internal__case__prod__def,axiom,
    produc6289240061155251307od_a_b = produc611504906096598721od_a_b ).

% internal_case_prod_def
thf(fact_263_internal__case__prod__def,axiom,
    produc987239214144727667_b_a_o = produc2053338206788791453_b_a_o ).

% internal_case_prod_def
thf(fact_264_internal__case__prod__def,axiom,
    produc2727628064737871481_a_b_a = produc543687385667932879_a_b_a ).

% internal_case_prod_def
thf(fact_265_pred__equals__eq2,axiom,
    ! [R: set_Pr7659163277375235966_b_a_a,S: set_Pr7659163277375235966_b_a_a] :
      ( ( ( ^ [X4: a,Y4: mappin4996136453398522101_b_a_a] : ( member7689197028375331655_b_a_a @ ( produc4259011374121840526_b_a_a @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: a,Y4: mappin4996136453398522101_b_a_a] : ( member7689197028375331655_b_a_a @ ( produc4259011374121840526_b_a_a @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_266_pred__equals__eq2,axiom,
    ! [R: set_Pr8515993315433149983_d_nat,S: set_Pr8515993315433149983_d_nat] :
      ( ( ( ^ [X4: a,Y4: list_P6684558232516339260_d_nat] : ( member8047447634869378944_d_nat @ ( produc919746265332163299_d_nat @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: a,Y4: list_P6684558232516339260_d_nat] : ( member8047447634869378944_d_nat @ ( produc919746265332163299_d_nat @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_267_pred__equals__eq2,axiom,
    ! [R: set_Pr5765251437205137415_a_b_a,S: set_Pr5765251437205137415_a_b_a] :
      ( ( ( ^ [X4: a,Y4: list_P2992919015626629540_a_b_a] : ( member378967531903691624_a_b_a @ ( produc6994869176599099851_a_b_a @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: a,Y4: list_P2992919015626629540_a_b_a] : ( member378967531903691624_a_b_a @ ( produc6994869176599099851_a_b_a @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_268_pred__equals__eq2,axiom,
    ! [R: set_Pr7142564897319301815od_a_b,S: set_Pr7142564897319301815od_a_b] :
      ( ( ( ^ [X4: a,Y4: list_P1467974523212632366od_a_b] : ( member8745141763671436032od_a_b @ ( produc6643264353547226055od_a_b @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: a,Y4: list_P1467974523212632366od_a_b] : ( member8745141763671436032od_a_b @ ( produc6643264353547226055od_a_b @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_269_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( ( ^ [X4: a,Y4: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: a,Y4: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_270_pred__equals__eq2,axiom,
    ! [R: set_Pr8583615357880026082_d_nat,S: set_Pr8583615357880026082_d_nat] :
      ( ( ( ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_271_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_a_b,S: set_Product_prod_a_b] :
      ( ( ( ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_272_pred__equals__eq2,axiom,
    ! [R: set_Pr8226792695654363348_a_b_a,S: set_Pr8226792695654363348_a_b_a] :
      ( ( ( ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_273_pred__equals__eq2,axiom,
    ! [R: set_Pr2043227878544749668_a_b_a,S: set_Pr2043227878544749668_a_b_a] :
      ( ( ( ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_274_pred__equals__eq2,axiom,
    ! [R: set_Pr2084289749507318346_a_b_a,S: set_Pr2084289749507318346_a_b_a] :
      ( ( ( ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ R ) )
        = ( ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_275_internal__case__prod__conv,axiom,
    ! [C2: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > produc2062766750192416900_a_b_a,A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a] :
      ( ( produc8391920324087245981_a_b_a @ C2 @ ( produc2616710729536263484_a_b_a @ A @ B ) )
      = ( C2 @ A @ B ) ) ).

% internal_case_prod_conv
thf(fact_276_internal__case__prod__conv,axiom,
    ! [C2: a > produc5456268773499131775_d_nat > produc1990351791824763820_d_nat,A: a,B: produc5456268773499131775_d_nat] :
      ( ( produc3580122457448217769_d_nat @ C2 @ ( produc5865182755557626534_d_nat @ A @ B ) )
      = ( C2 @ A @ B ) ) ).

% internal_case_prod_conv
thf(fact_277_internal__case__prod__conv,axiom,
    ! [C2: a > produc5456268773499131775_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,A: a,B: produc5456268773499131775_d_nat] :
      ( ( produc4871049226530554055_a_b_a @ C2 @ ( produc5865182755557626534_d_nat @ A @ B ) )
      = ( C2 @ A @ B ) ) ).

% internal_case_prod_conv
thf(fact_278_internal__case__prod__conv,axiom,
    ! [C2: a > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,A: a,B: mappin2365780136655028071_a_b_a] :
      ( ( produc5628374565148576719_a_b_a @ C2 @ ( produc1583556749610718734_a_b_a @ A @ B ) )
      = ( C2 @ A @ B ) ) ).

% internal_case_prod_conv
thf(fact_279_internal__case__prod__conv,axiom,
    ! [C2: a > b > product_prod_a_b,A: a,B: b] :
      ( ( produc3797153600616317541od_a_b @ C2 @ ( product_Pair_a_b @ A @ B ) )
      = ( C2 @ A @ B ) ) ).

% internal_case_prod_conv
thf(fact_280_mapping__eqI,axiom,
    ! [M: mappin4996136453398522101_b_a_a,M2: mappin4996136453398522101_b_a_a] :
      ( ! [X3: produc7822699449245016212_a_b_a] :
          ( ( lookup1409243802988840364_b_a_a @ M @ X3 )
          = ( lookup1409243802988840364_b_a_a @ M2 @ X3 ) )
     => ( M = M2 ) ) ).

% mapping_eqI
thf(fact_281_mapping__eqI,axiom,
    ! [M: mappin187059948859696622_a_b_a,M2: mappin187059948859696622_a_b_a] :
      ( ! [X3: list_P6684558232516339260_d_nat] :
          ( ( lookup4661578797684577249_a_b_a @ M @ X3 )
          = ( lookup4661578797684577249_a_b_a @ M2 @ X3 ) )
     => ( M = M2 ) ) ).

% mapping_eqI
thf(fact_282_mapping__eqI,axiom,
    ! [M: mappin6535725686296717250_d_nat,M2: mappin6535725686296717250_d_nat] :
      ( ! [X3: a] :
          ( ( lookup8480750236328838337_d_nat @ M @ X3 )
          = ( lookup8480750236328838337_d_nat @ M2 @ X3 ) )
     => ( M = M2 ) ) ).

% mapping_eqI
thf(fact_283_mapping__eqI,axiom,
    ! [M: mappin4519906018992269226_a_b_a,M2: mappin4519906018992269226_a_b_a] :
      ( ! [X3: a] :
          ( ( lookup1261341372242361769_a_b_a @ M @ X3 )
          = ( lookup1261341372242361769_a_b_a @ M2 @ X3 ) )
     => ( M = M2 ) ) ).

% mapping_eqI
thf(fact_284_mapping__eqI,axiom,
    ! [M: mapping_a_b,M2: mapping_a_b] :
      ( ! [X3: a] :
          ( ( lookup_a_b @ M @ X3 )
          = ( lookup_a_b @ M2 @ X3 ) )
     => ( M = M2 ) ) ).

% mapping_eqI
thf(fact_285_mapping__eqI,axiom,
    ! [M: mappin2365780136655028071_a_b_a,M2: mappin2365780136655028071_a_b_a] :
      ( ! [X3: product_prod_a_b] :
          ( ( lookup863726980359885586_a_b_a @ M @ X3 )
          = ( lookup863726980359885586_a_b_a @ M2 @ X3 ) )
     => ( M = M2 ) ) ).

% mapping_eqI
thf(fact_286_curry__case__prod,axiom,
    ! [F: a > b > product_prod_a_b] :
      ( ( produc3755105452235003484od_a_b @ ( produc8992199381948149691od_a_b @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_287_curry__case__prod,axiom,
    ! [F: a > produc5456268773499131775_d_nat > produc1990351791824763820_d_nat] :
      ( ( produc8282818114755544864_d_nat @ ( produc5272039204407661311_d_nat @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_288_curry__case__prod,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a] :
      ( ( produc6137926072827258392_a_b_a @ ( produc452744236373541113_a_b_a @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_289_curry__case__prod,axiom,
    ! [F: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a] :
      ( ( produc9090429389446707860_a_b_a @ ( produc5613835436409813491_a_b_a @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_290_curry__case__prod,axiom,
    ! [F: a > produc5456268773499131775_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a] :
      ( ( produc5534703194321516048_a_b_a @ ( produc5638855421663032305_a_b_a @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_291_curry__case__prod,axiom,
    ! [F: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > produc2062766750192416900_a_b_a] :
      ( ( produc1013893662665017236_a_b_a @ ( produc3256888519411912691_a_b_a @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_292_curry__case__prod,axiom,
    ! [F: a > b > $o] :
      ( ( product_curry_a_b_o @ ( produc3537405659489547051_a_b_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_293_curry__case__prod,axiom,
    ! [F: a > b > list_P1467974523212632366od_a_b] :
      ( ( produc6509513668519493858od_a_b @ ( produc611504906096598721od_a_b @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_294_curry__case__prod,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > $o] :
      ( ( produc3842274730932565308_b_a_o @ ( produc2053338206788791453_b_a_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_295_curry__case__prod,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > produc7822699449245016212_a_b_a] :
      ( ( produc4095722223507068912_a_b_a @ ( produc543687385667932879_a_b_a @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_296_case__prod__curry,axiom,
    ! [F: product_prod_a_b > product_prod_a_b] :
      ( ( produc8992199381948149691od_a_b @ ( produc3755105452235003484od_a_b @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_297_case__prod__curry,axiom,
    ! [F: produc1990351791824763820_d_nat > produc1990351791824763820_d_nat] :
      ( ( produc5272039204407661311_d_nat @ ( produc8282818114755544864_d_nat @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_298_case__prod__curry,axiom,
    ! [F: produc7822699449245016212_a_b_a > produc6343861673719254644_a_b_a] :
      ( ( produc452744236373541113_a_b_a @ ( produc6137926072827258392_a_b_a @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_299_case__prod__curry,axiom,
    ! [F: produc6343861673719254644_a_b_a > produc6343861673719254644_a_b_a] :
      ( ( produc5613835436409813491_a_b_a @ ( produc9090429389446707860_a_b_a @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_300_case__prod__curry,axiom,
    ! [F: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a] :
      ( ( produc5638855421663032305_a_b_a @ ( produc5534703194321516048_a_b_a @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_301_case__prod__curry,axiom,
    ! [F: produc2062766750192416900_a_b_a > produc2062766750192416900_a_b_a] :
      ( ( produc3256888519411912691_a_b_a @ ( produc1013893662665017236_a_b_a @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_302_case__prod__curry,axiom,
    ! [F: product_prod_a_b > $o] :
      ( ( produc3537405659489547051_a_b_o @ ( product_curry_a_b_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_303_case__prod__curry,axiom,
    ! [F: product_prod_a_b > list_P1467974523212632366od_a_b] :
      ( ( produc611504906096598721od_a_b @ ( produc6509513668519493858od_a_b @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_304_case__prod__curry,axiom,
    ! [F: produc7822699449245016212_a_b_a > $o] :
      ( ( produc2053338206788791453_b_a_o @ ( produc3842274730932565308_b_a_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_305_case__prod__curry,axiom,
    ! [F: produc7822699449245016212_a_b_a > produc7822699449245016212_a_b_a] :
      ( ( produc543687385667932879_a_b_a @ ( produc4095722223507068912_a_b_a @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_306_curryI,axiom,
    ! [F: produc3619439354540788382_b_a_a > $o,A: a,B: mappin4996136453398522101_b_a_a] :
      ( ( F @ ( produc4259011374121840526_b_a_a @ A @ B ) )
     => ( produc2925195325953240534_a_a_o @ F @ A @ B ) ) ).

% curryI
thf(fact_307_curryI,axiom,
    ! [F: produc9176665399011442921_d_nat > $o,A: a,B: list_P6684558232516339260_d_nat] :
      ( ( F @ ( produc919746265332163299_d_nat @ A @ B ) )
     => ( produc8694939645206489895_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_308_curryI,axiom,
    ! [F: produc7083967959564351825_a_b_a > $o,A: a,B: list_P2992919015626629540_a_b_a] :
      ( ( F @ ( produc6994869176599099851_a_b_a @ A @ B ) )
     => ( produc3597518269466942399_b_a_o @ F @ A @ B ) ) ).

% curryI
thf(fact_309_curryI,axiom,
    ! [F: produc2639161875728825943od_a_b > $o,A: a,B: list_P1467974523212632366od_a_b] :
      ( ( F @ ( produc6643264353547226055od_a_b @ A @ B ) )
     => ( produc1240654420938582365_a_b_o @ F @ A @ B ) ) ).

% curryI
thf(fact_310_curryI,axiom,
    ! [F: product_prod_a_a > $o,A: a,B: a] :
      ( ( F @ ( product_Pair_a_a @ A @ B ) )
     => ( product_curry_a_a_o @ F @ A @ B ) ) ).

% curryI
thf(fact_311_curryI,axiom,
    ! [F: produc1990351791824763820_d_nat > $o,A: a,B: produc5456268773499131775_d_nat] :
      ( ( F @ ( produc5865182755557626534_d_nat @ A @ B ) )
     => ( produc2404724602700222756_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_312_curryI,axiom,
    ! [F: product_prod_a_b > $o,A: a,B: b] :
      ( ( F @ ( product_Pair_a_b @ A @ B ) )
     => ( product_curry_a_b_o @ F @ A @ B ) ) ).

% curryI
thf(fact_313_curryI,axiom,
    ! [F: produc6343861673719254644_a_b_a > $o,A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a] :
      ( ( F @ ( produc1681437058057156268_a_b_a @ A @ B ) )
     => ( produc8543931719223760824_b_a_o @ F @ A @ B ) ) ).

% curryI
thf(fact_314_curryI,axiom,
    ! [F: produc2062766750192416900_a_b_a > $o,A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a] :
      ( ( F @ ( produc2616710729536263484_a_b_a @ A @ B ) )
     => ( produc2006925953541372584_b_a_o @ F @ A @ B ) ) ).

% curryI
thf(fact_315_curryI,axiom,
    ! [F: produc7822699449245016212_a_b_a > $o,A: a,B: mappin2365780136655028071_a_b_a] :
      ( ( F @ ( produc1583556749610718734_a_b_a @ A @ B ) )
     => ( produc3842274730932565308_b_a_o @ F @ A @ B ) ) ).

% curryI
thf(fact_316_split__part,axiom,
    ! [P: $o,Q2: a > mappin2365780136655028071_a_b_a > $o] :
      ( ( produc2053338206788791453_b_a_o
        @ ^ [A5: a,B4: mappin2365780136655028071_a_b_a] :
            ( P
            & ( Q2 @ A5 @ B4 ) ) )
      = ( ^ [Ab: produc7822699449245016212_a_b_a] :
            ( P
            & ( produc2053338206788791453_b_a_o @ Q2 @ Ab ) ) ) ) ).

% split_part
thf(fact_317_split__part,axiom,
    ! [P: $o,Q2: a > b > $o] :
      ( ( produc3537405659489547051_a_b_o
        @ ^ [A5: a,B4: b] :
            ( P
            & ( Q2 @ A5 @ B4 ) ) )
      = ( ^ [Ab: product_prod_a_b] :
            ( P
            & ( produc3537405659489547051_a_b_o @ Q2 @ Ab ) ) ) ) ).

% split_part
thf(fact_318_curry__conv,axiom,
    ( produc5534703194321516048_a_b_a
    = ( ^ [F2: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,A5: a,B4: produc5456268773499131775_d_nat] : ( F2 @ ( produc5865182755557626534_d_nat @ A5 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_319_curry__conv,axiom,
    ( produc6137926072827258392_a_b_a
    = ( ^ [F2: produc7822699449245016212_a_b_a > produc6343861673719254644_a_b_a,A5: a,B4: mappin2365780136655028071_a_b_a] : ( F2 @ ( produc1583556749610718734_a_b_a @ A5 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_320_curry__conv,axiom,
    ( produc3842274730932565308_b_a_o
    = ( ^ [F2: produc7822699449245016212_a_b_a > $o,A5: a,B4: mappin2365780136655028071_a_b_a] : ( F2 @ ( produc1583556749610718734_a_b_a @ A5 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_321_curry__conv,axiom,
    ( produc3755105452235003484od_a_b
    = ( ^ [F2: product_prod_a_b > product_prod_a_b,A5: a,B4: b] : ( F2 @ ( product_Pair_a_b @ A5 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_322_curry__conv,axiom,
    ( product_curry_a_b_o
    = ( ^ [F2: product_prod_a_b > $o,A5: a,B4: b] : ( F2 @ ( product_Pair_a_b @ A5 @ B4 ) ) ) ) ).

% curry_conv
thf(fact_323_prod_Odisc__eq__case,axiom,
    ! [Prod: produc7822699449245016212_a_b_a] :
      ( produc2053338206788791453_b_a_o
      @ ^ [Uu: a,Uv: mappin2365780136655028071_a_b_a] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_324_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_a_b] :
      ( produc3537405659489547051_a_b_o
      @ ^ [Uu: a,Uv: b] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_325_curry__K,axiom,
    ! [C2: mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a] :
      ( ( produc5534703194321516048_a_b_a
        @ ^ [X4: produc1990351791824763820_d_nat] : C2 )
      = ( ^ [X4: a,Y4: produc5456268773499131775_d_nat] : C2 ) ) ).

% curry_K
thf(fact_326_curry__K,axiom,
    ! [C2: produc6343861673719254644_a_b_a] :
      ( ( produc6137926072827258392_a_b_a
        @ ^ [X4: produc7822699449245016212_a_b_a] : C2 )
      = ( ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : C2 ) ) ).

% curry_K
thf(fact_327_curry__K,axiom,
    ! [C2: $o] :
      ( ( produc3842274730932565308_b_a_o
        @ ^ [X4: produc7822699449245016212_a_b_a] : C2 )
      = ( ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : C2 ) ) ).

% curry_K
thf(fact_328_curry__K,axiom,
    ! [C2: product_prod_a_b] :
      ( ( produc3755105452235003484od_a_b
        @ ^ [X4: product_prod_a_b] : C2 )
      = ( ^ [X4: a,Y4: b] : C2 ) ) ).

% curry_K
thf(fact_329_curry__K,axiom,
    ! [C2: $o] :
      ( ( product_curry_a_b_o
        @ ^ [X4: product_prod_a_b] : C2 )
      = ( ^ [X4: a,Y4: b] : C2 ) ) ).

% curry_K
thf(fact_330_curryD,axiom,
    ! [F: produc3619439354540788382_b_a_a > $o,A: a,B: mappin4996136453398522101_b_a_a] :
      ( ( produc2925195325953240534_a_a_o @ F @ A @ B )
     => ( F @ ( produc4259011374121840526_b_a_a @ A @ B ) ) ) ).

% curryD
thf(fact_331_curryD,axiom,
    ! [F: produc9176665399011442921_d_nat > $o,A: a,B: list_P6684558232516339260_d_nat] :
      ( ( produc8694939645206489895_nat_o @ F @ A @ B )
     => ( F @ ( produc919746265332163299_d_nat @ A @ B ) ) ) ).

% curryD
thf(fact_332_curryD,axiom,
    ! [F: produc7083967959564351825_a_b_a > $o,A: a,B: list_P2992919015626629540_a_b_a] :
      ( ( produc3597518269466942399_b_a_o @ F @ A @ B )
     => ( F @ ( produc6994869176599099851_a_b_a @ A @ B ) ) ) ).

% curryD
thf(fact_333_curryD,axiom,
    ! [F: produc2639161875728825943od_a_b > $o,A: a,B: list_P1467974523212632366od_a_b] :
      ( ( produc1240654420938582365_a_b_o @ F @ A @ B )
     => ( F @ ( produc6643264353547226055od_a_b @ A @ B ) ) ) ).

% curryD
thf(fact_334_curryD,axiom,
    ! [F: product_prod_a_a > $o,A: a,B: a] :
      ( ( product_curry_a_a_o @ F @ A @ B )
     => ( F @ ( product_Pair_a_a @ A @ B ) ) ) ).

% curryD
thf(fact_335_curryD,axiom,
    ! [F: produc1990351791824763820_d_nat > $o,A: a,B: produc5456268773499131775_d_nat] :
      ( ( produc2404724602700222756_nat_o @ F @ A @ B )
     => ( F @ ( produc5865182755557626534_d_nat @ A @ B ) ) ) ).

% curryD
thf(fact_336_curryD,axiom,
    ! [F: product_prod_a_b > $o,A: a,B: b] :
      ( ( product_curry_a_b_o @ F @ A @ B )
     => ( F @ ( product_Pair_a_b @ A @ B ) ) ) ).

% curryD
thf(fact_337_curryD,axiom,
    ! [F: produc6343861673719254644_a_b_a > $o,A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a] :
      ( ( produc8543931719223760824_b_a_o @ F @ A @ B )
     => ( F @ ( produc1681437058057156268_a_b_a @ A @ B ) ) ) ).

% curryD
thf(fact_338_curryD,axiom,
    ! [F: produc2062766750192416900_a_b_a > $o,A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a] :
      ( ( produc2006925953541372584_b_a_o @ F @ A @ B )
     => ( F @ ( produc2616710729536263484_a_b_a @ A @ B ) ) ) ).

% curryD
thf(fact_339_curryD,axiom,
    ! [F: produc7822699449245016212_a_b_a > $o,A: a,B: mappin2365780136655028071_a_b_a] :
      ( ( produc3842274730932565308_b_a_o @ F @ A @ B )
     => ( F @ ( produc1583556749610718734_a_b_a @ A @ B ) ) ) ).

% curryD
thf(fact_340_curryE,axiom,
    ! [F: produc3619439354540788382_b_a_a > $o,A: a,B: mappin4996136453398522101_b_a_a] :
      ( ( produc2925195325953240534_a_a_o @ F @ A @ B )
     => ( F @ ( produc4259011374121840526_b_a_a @ A @ B ) ) ) ).

% curryE
thf(fact_341_curryE,axiom,
    ! [F: produc9176665399011442921_d_nat > $o,A: a,B: list_P6684558232516339260_d_nat] :
      ( ( produc8694939645206489895_nat_o @ F @ A @ B )
     => ( F @ ( produc919746265332163299_d_nat @ A @ B ) ) ) ).

% curryE
thf(fact_342_curryE,axiom,
    ! [F: produc7083967959564351825_a_b_a > $o,A: a,B: list_P2992919015626629540_a_b_a] :
      ( ( produc3597518269466942399_b_a_o @ F @ A @ B )
     => ( F @ ( produc6994869176599099851_a_b_a @ A @ B ) ) ) ).

% curryE
thf(fact_343_curryE,axiom,
    ! [F: produc2639161875728825943od_a_b > $o,A: a,B: list_P1467974523212632366od_a_b] :
      ( ( produc1240654420938582365_a_b_o @ F @ A @ B )
     => ( F @ ( produc6643264353547226055od_a_b @ A @ B ) ) ) ).

% curryE
thf(fact_344_curryE,axiom,
    ! [F: product_prod_a_a > $o,A: a,B: a] :
      ( ( product_curry_a_a_o @ F @ A @ B )
     => ( F @ ( product_Pair_a_a @ A @ B ) ) ) ).

% curryE
thf(fact_345_curryE,axiom,
    ! [F: produc1990351791824763820_d_nat > $o,A: a,B: produc5456268773499131775_d_nat] :
      ( ( produc2404724602700222756_nat_o @ F @ A @ B )
     => ( F @ ( produc5865182755557626534_d_nat @ A @ B ) ) ) ).

% curryE
thf(fact_346_curryE,axiom,
    ! [F: product_prod_a_b > $o,A: a,B: b] :
      ( ( product_curry_a_b_o @ F @ A @ B )
     => ( F @ ( product_Pair_a_b @ A @ B ) ) ) ).

% curryE
thf(fact_347_curryE,axiom,
    ! [F: produc6343861673719254644_a_b_a > $o,A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a] :
      ( ( produc8543931719223760824_b_a_o @ F @ A @ B )
     => ( F @ ( produc1681437058057156268_a_b_a @ A @ B ) ) ) ).

% curryE
thf(fact_348_curryE,axiom,
    ! [F: produc2062766750192416900_a_b_a > $o,A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a] :
      ( ( produc2006925953541372584_b_a_o @ F @ A @ B )
     => ( F @ ( produc2616710729536263484_a_b_a @ A @ B ) ) ) ).

% curryE
thf(fact_349_curryE,axiom,
    ! [F: produc7822699449245016212_a_b_a > $o,A: a,B: mappin2365780136655028071_a_b_a] :
      ( ( produc3842274730932565308_b_a_o @ F @ A @ B )
     => ( F @ ( produc1583556749610718734_a_b_a @ A @ B ) ) ) ).

% curryE
thf(fact_350_curry__def,axiom,
    ( produc5534703194321516048_a_b_a
    = ( ^ [C3: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a,X4: a,Y4: produc5456268773499131775_d_nat] : ( C3 @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) ) ) ) ).

% curry_def
thf(fact_351_curry__def,axiom,
    ( produc6137926072827258392_a_b_a
    = ( ^ [C3: produc7822699449245016212_a_b_a > produc6343861673719254644_a_b_a,X4: a,Y4: mappin2365780136655028071_a_b_a] : ( C3 @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) ) ) ) ).

% curry_def
thf(fact_352_curry__def,axiom,
    ( produc3842274730932565308_b_a_o
    = ( ^ [C3: produc7822699449245016212_a_b_a > $o,X4: a,Y4: mappin2365780136655028071_a_b_a] : ( C3 @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) ) ) ) ).

% curry_def
thf(fact_353_curry__def,axiom,
    ( produc3755105452235003484od_a_b
    = ( ^ [C3: product_prod_a_b > product_prod_a_b,X4: a,Y4: b] : ( C3 @ ( product_Pair_a_b @ X4 @ Y4 ) ) ) ) ).

% curry_def
thf(fact_354_curry__def,axiom,
    ( product_curry_a_b_o
    = ( ^ [C3: product_prod_a_b > $o,X4: a,Y4: b] : ( C3 @ ( product_Pair_a_b @ X4 @ Y4 ) ) ) ) ).

% curry_def
thf(fact_355_cstep__def,axiom,
    ( cstep_9086493308577808298_b_a_a
    = ( ^ [Step: a > mappin4996136453398522101_b_a_a > a,St: mappin17023578605762161_a_a_a,Q4: a,Bs2: mappin4996136453398522101_b_a_a] :
          ( case_o6478378332473707667_a_a_a @ ( produc4266190156793545368_a_a_a @ ( Step @ Q4 @ Bs2 ) @ ( update2949569333909129419_a_a_a @ ( produc4259011374121840526_b_a_a @ Q4 @ Bs2 ) @ ( Step @ Q4 @ Bs2 ) @ St ) )
          @ ^ [V: a] : ( produc4266190156793545368_a_a_a @ V @ St )
          @ ( lookup5009658751286874908_a_a_a @ St @ ( produc4259011374121840526_b_a_a @ Q4 @ Bs2 ) ) ) ) ) ).

% cstep_def
thf(fact_356_cstep__def,axiom,
    ( cstep_6750614933438834119_d_nat
    = ( ^ [Step: a > list_P6684558232516339260_d_nat > a,St: mappin6510584212744718816_nat_a,Q4: a,Bs2: list_P6684558232516339260_d_nat] :
          ( case_o8942390679909280704at_a_a @ ( produc185989309827705465_nat_a @ ( Step @ Q4 @ Bs2 ) @ ( update8892332461331997608_nat_a @ ( produc919746265332163299_d_nat @ Q4 @ Bs2 ) @ ( Step @ Q4 @ Bs2 ) @ St ) )
          @ ^ [V: a] : ( produc185989309827705465_nat_a @ V @ St )
          @ ( lookup6857919033935848087_nat_a @ St @ ( produc919746265332163299_d_nat @ Q4 @ Bs2 ) ) ) ) ) ).

% cstep_def
thf(fact_357_cstep__def,axiom,
    ( cstep_3906380498826956463_a_b_a
    = ( ^ [Step: a > list_P2992919015626629540_a_b_a > a,St: mappin550309913600311800_b_a_a,Q4: a,Bs2: list_P2992919015626629540_a_b_a] :
          ( case_o3814117949224516776_a_a_a @ ( produc1160361938620327697_b_a_a @ ( Step @ Q4 @ Bs2 ) @ ( update556944065238765376_b_a_a @ ( produc6994869176599099851_a_b_a @ Q4 @ Bs2 ) @ ( Step @ Q4 @ Bs2 ) @ St ) )
          @ ^ [V: a] : ( produc1160361938620327697_b_a_a @ V @ St )
          @ ( lookup1643283453670772143_b_a_a @ St @ ( produc6994869176599099851_a_b_a @ Q4 @ Bs2 ) ) ) ) ) ).

% cstep_def
thf(fact_358_cstep__def,axiom,
    ( cstep_8261300269631392739od_a_b
    = ( ^ [Step: a > list_P1467974523212632366od_a_b > a,St: mappin1995428302098636536_a_b_a,Q4: a,Bs2: list_P1467974523212632366od_a_b] :
          ( case_o2082071590584463180_b_a_a @ ( produc5997378463071106847_a_b_a @ ( Step @ Q4 @ Bs2 ) @ ( update3671798976407283026_a_b_a @ ( produc6643264353547226055od_a_b @ Q4 @ Bs2 ) @ ( Step @ Q4 @ Bs2 ) @ St ) )
          @ ^ [V: a] : ( produc5997378463071106847_a_b_a @ V @ St )
          @ ( lookup5884355714128708515_a_b_a @ St @ ( produc6643264353547226055od_a_b @ Q4 @ Bs2 ) ) ) ) ) ).

% cstep_def
thf(fact_359_cstep__def,axiom,
    ( cstep_a_a
    = ( ^ [Step: a > a > a,St: mappin3001429137260014760_a_a_a,Q4: a,Bs2: a] :
          ( case_o8071424220481837212_a_a_a @ ( produc2219205750215705423_a_a_a @ ( Step @ Q4 @ Bs2 ) @ ( update1387224295439371394_a_a_a @ ( product_Pair_a_a @ Q4 @ Bs2 ) @ ( Step @ Q4 @ Bs2 ) @ St ) )
          @ ^ [V: a] : ( produc2219205750215705423_a_a_a @ V @ St )
          @ ( lookup1499375980964872275_a_a_a @ St @ ( product_Pair_a_a @ Q4 @ Bs2 ) ) ) ) ) ).

% cstep_def
thf(fact_360_cstep__def,axiom,
    ( cstep_4766452580563721098_d_nat
    = ( ^ [Step: a > produc5456268773499131775_d_nat > a,St: mappin2367819674035305821_nat_a,Q4: a,Bs2: produc5456268773499131775_d_nat] :
          ( case_o4872500525950110979at_a_a @ ( produc1686585632564395894_nat_a @ ( Step @ Q4 @ Bs2 ) @ ( update6875199776339664293_nat_a @ ( produc5865182755557626534_d_nat @ Q4 @ Bs2 ) @ ( Step @ Q4 @ Bs2 ) @ St ) )
          @ ^ [V: a] : ( produc1686585632564395894_nat_a @ V @ St )
          @ ( lookup333719581650085652_nat_a @ St @ ( produc5865182755557626534_d_nat @ Q4 @ Bs2 ) ) ) ) ) ).

% cstep_def
thf(fact_361_cstep__def,axiom,
    ( cstep_9201453730389171400_a_b_a
    = ( ^ [Step: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc1990351791824763820_d_nat,St: mappin182839930001496549_d_nat,Q4: produc1990351791824763820_d_nat,Bs2: mappin2365780136655028071_a_b_a] :
          ( case_o7371686905944619111_d_nat @ ( produc7378667233664236768_d_nat @ ( Step @ Q4 @ Bs2 ) @ ( update8467124912708031989_d_nat @ ( produc1681437058057156268_a_b_a @ Q4 @ Bs2 ) @ ( Step @ Q4 @ Bs2 ) @ St ) )
          @ ^ [V: produc1990351791824763820_d_nat] : ( produc7378667233664236768_d_nat @ V @ St )
          @ ( lookup1478974430281096036_d_nat @ St @ ( produc1681437058057156268_a_b_a @ Q4 @ Bs2 ) ) ) ) ) ).

% cstep_def
thf(fact_362_cstep__def,axiom,
    ( cstep_8854914602869980504_a_b_a
    = ( ^ [Step: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > list_P6684558232516339260_d_nat,St: mappin505876631851820293_d_nat,Q4: list_P6684558232516339260_d_nat,Bs2: mappin2365780136655028071_a_b_a] :
          ( case_o991627342249586311_d_nat @ ( produc6939990982589020784_d_nat @ ( Step @ Q4 @ Bs2 ) @ ( update7531612778948884501_d_nat @ ( produc2616710729536263484_a_b_a @ Q4 @ Bs2 ) @ ( Step @ Q4 @ Bs2 ) @ St ) )
          @ ^ [V: list_P6684558232516339260_d_nat] : ( produc6939990982589020784_d_nat @ V @ St )
          @ ( lookup2178750111427327620_d_nat @ St @ ( produc2616710729536263484_a_b_a @ Q4 @ Bs2 ) ) ) ) ) ).

% cstep_def
thf(fact_363_cstep__def,axiom,
    ( cstep_6646870280633757938_a_b_a
    = ( ^ [Step: a > mappin2365780136655028071_a_b_a > a,St: mappin4996136453398522101_b_a_a,Q4: a,Bs2: mappin2365780136655028071_a_b_a] :
          ( case_o8746674778622544491_a_a_a @ ( produc4259011374121840526_b_a_a @ ( Step @ Q4 @ Bs2 ) @ ( update5685058447907601597_b_a_a @ ( produc1583556749610718734_a_b_a @ Q4 @ Bs2 ) @ ( Step @ Q4 @ Bs2 ) @ St ) )
          @ ^ [V: a] : ( produc4259011374121840526_b_a_a @ V @ St )
          @ ( lookup1409243802988840364_b_a_a @ St @ ( produc1583556749610718734_a_b_a @ Q4 @ Bs2 ) ) ) ) ) ).

% cstep_def
thf(fact_364_cstep__def,axiom,
    ( cstep_a_b
    = ( ^ [Step: a > b > a,St: mappin2365780136655028071_a_b_a,Q4: a,Bs2: b] :
          ( case_o140846598303662877_b_a_a @ ( produc1583556749610718734_a_b_a @ ( Step @ Q4 @ Bs2 ) @ ( update751575294834384705_a_b_a @ ( product_Pair_a_b @ Q4 @ Bs2 ) @ ( Step @ Q4 @ Bs2 ) @ St ) )
          @ ^ [V: a] : ( produc1583556749610718734_a_b_a @ V @ St )
          @ ( lookup863726980359885586_a_b_a @ St @ ( product_Pair_a_b @ Q4 @ Bs2 ) ) ) ) ) ).

% cstep_def
thf(fact_365_All__mapping__def,axiom,
    ( all_ma3172627230583229140_a_b_a
    = ( ^ [M3: mappin187059948859696622_a_b_a,P5: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > $o] :
        ! [X4: list_P6684558232516339260_d_nat] : ( case_o630936899899582644_a_b_a @ $true @ ( P5 @ X4 ) @ ( lookup4661578797684577249_a_b_a @ M3 @ X4 ) ) ) ) ).

% All_mapping_def
thf(fact_366_All__mapping__def,axiom,
    ( all_ma1625059968680912654_d_nat
    = ( ^ [M3: mappin6535725686296717250_d_nat,P5: a > produc5456268773499131775_d_nat > $o] :
        ! [X4: a] : ( case_o6014610241001325132_d_nat @ $true @ ( P5 @ X4 ) @ ( lookup8480750236328838337_d_nat @ M3 @ X4 ) ) ) ) ).

% All_mapping_def
thf(fact_367_All__mapping__def,axiom,
    ( all_ma2913212136517639798_a_b_a
    = ( ^ [M3: mappin4519906018992269226_a_b_a,P5: a > mappin2365780136655028071_a_b_a > $o] :
        ! [X4: a] : ( case_o630936899899582644_a_b_a @ $true @ ( P5 @ X4 ) @ ( lookup1261341372242361769_a_b_a @ M3 @ X4 ) ) ) ) ).

% All_mapping_def
thf(fact_368_All__mapping__def,axiom,
    ( all_mapping_a_b
    = ( ^ [M3: mapping_a_b,P5: a > b > $o] :
        ! [X4: a] : ( case_option_o_b @ $true @ ( P5 @ X4 ) @ ( lookup_a_b @ M3 @ X4 ) ) ) ) ).

% All_mapping_def
thf(fact_369_All__mapping__def,axiom,
    ( all_ma7748622604349067641_b_a_a
    = ( ^ [M3: mappin4996136453398522101_b_a_a,P5: produc7822699449245016212_a_b_a > a > $o] :
        ! [X4: produc7822699449245016212_a_b_a] : ( case_option_o_a @ $true @ ( P5 @ X4 ) @ ( lookup1409243802988840364_b_a_a @ M3 @ X4 ) ) ) ) ).

% All_mapping_def
thf(fact_370_All__mapping__def,axiom,
    ( all_ma2923619520195476869_a_b_a
    = ( ^ [M3: mappin2365780136655028071_a_b_a,P5: product_prod_a_b > a > $o] :
        ! [X4: product_prod_a_b] : ( case_option_o_a @ $true @ ( P5 @ X4 ) @ ( lookup863726980359885586_a_b_a @ M3 @ X4 ) ) ) ) ).

% All_mapping_def
thf(fact_371_lookup__default__def,axiom,
    ( lookup1777753518785040938_d_nat
    = ( ^ [D: mappin2365780136655028071_a_b_a,M3: mappin187059948859696622_a_b_a,K: list_P6684558232516339260_d_nat] :
          ( case_o7677175464458025051_a_b_a @ D
          @ ^ [V: mappin2365780136655028071_a_b_a] : V
          @ ( lookup4661578797684577249_a_b_a @ M3 @ K ) ) ) ) ).

% lookup_default_def
thf(fact_372_lookup__default__def,axiom,
    ( lookup2689911013986210168_nat_a
    = ( ^ [D: produc5456268773499131775_d_nat,M3: mappin6535725686296717250_d_nat,K: a] :
          ( case_o7522508894794884235_d_nat @ D
          @ ^ [V: produc5456268773499131775_d_nat] : V
          @ ( lookup8480750236328838337_d_nat @ M3 @ K ) ) ) ) ).

% lookup_default_def
thf(fact_373_lookup__default__def,axiom,
    ( lookup4260590398720651408_b_a_a
    = ( ^ [D: mappin2365780136655028071_a_b_a,M3: mappin4519906018992269226_a_b_a,K: a] :
          ( case_o7677175464458025051_a_b_a @ D
          @ ^ [V: mappin2365780136655028071_a_b_a] : V
          @ ( lookup1261341372242361769_a_b_a @ M3 @ K ) ) ) ) ).

% lookup_default_def
thf(fact_374_lookup__default__def,axiom,
    ( lookup_default_b_a
    = ( ^ [D: b,M3: mapping_a_b,K: a] :
          ( case_option_b_b @ D
          @ ^ [V: b] : V
          @ ( lookup_a_b @ M3 @ K ) ) ) ) ).

% lookup_default_def
thf(fact_375_lookup__default__def,axiom,
    ( lookup4806362731325673933_a_b_a
    = ( ^ [D: a,M3: mappin4996136453398522101_b_a_a,K: produc7822699449245016212_a_b_a] :
          ( case_option_a_a @ D
          @ ^ [V: a] : V
          @ ( lookup1409243802988840364_b_a_a @ M3 @ K ) ) ) ) ).

% lookup_default_def
thf(fact_376_lookup__default__def,axiom,
    ( lookup592102128023449775od_a_b
    = ( ^ [D: a,M3: mappin2365780136655028071_a_b_a,K: product_prod_a_b] :
          ( case_option_a_a @ D
          @ ^ [V: a] : V
          @ ( lookup863726980359885586_a_b_a @ M3 @ K ) ) ) ) ).

% lookup_default_def
thf(fact_377_same__fst__def,axiom,
    ( same_f2554954299287463569_a_b_a
    = ( ^ [P5: list_P6684558232516339260_d_nat > $o,R4: list_P6684558232516339260_d_nat > set_Pr1410444956374649615_a_b_a] :
          ( collec7317118955271278738_a_b_a
          @ ( produc8109344284017171820_b_a_o
            @ ( produc7322125808614349514_b_a_o
              @ ^ [X5: list_P6684558232516339260_d_nat,Y6: mappin2365780136655028071_a_b_a] :
                  ( produc6164275942970966535_b_a_o
                  @ ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] :
                      ( ( X5 = X4 )
                      & ( P5 @ X4 )
                      & ( member1544477001419906904_a_b_a @ ( produc5677138682576874855_a_b_a @ Y6 @ Y4 ) @ ( R4 @ X4 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_378_same__fst__def,axiom,
    ( same_f8862508556523255425_a_b_a
    = ( ^ [P5: produc1990351791824763820_d_nat > $o,R4: produc1990351791824763820_d_nat > set_Pr1410444956374649615_a_b_a] :
          ( collec1164995411054953490_a_b_a
          @ ( produc5089158207452318700_b_a_o
            @ ( produc708482854371332170_b_a_o
              @ ^ [X5: produc1990351791824763820_d_nat,Y6: mappin2365780136655028071_a_b_a] :
                  ( produc6407700812349938327_b_a_o
                  @ ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] :
                      ( ( X5 = X4 )
                      & ( P5 @ X4 )
                      & ( member1544477001419906904_a_b_a @ ( produc5677138682576874855_a_b_a @ Y6 @ Y4 ) @ ( R4 @ X4 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_379_same__fst__def,axiom,
    ( same_f3903536699954488849_d_nat
    = ( ^ [P5: a > $o,R4: a > set_Pr2853997496678843199_d_nat] :
          ( collec2880425841902469156_d_nat
          @ ( produc4563981736279841938_nat_o
            @ ( produc6235630094606560144_nat_o
              @ ^ [X5: a,Y6: produc5456268773499131775_d_nat] :
                  ( produc1905287380190322693_nat_o
                  @ ^ [X4: a,Y4: produc5456268773499131775_d_nat] :
                      ( ( X5 = X4 )
                      & ( P5 @ X4 )
                      & ( member4240823983429516168_d_nat @ ( produc3096269196918525335_d_nat @ Y6 @ Y4 ) @ ( R4 @ X4 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_380_same__fst__def,axiom,
    ( same_f5330524035721250553_a_b_a
    = ( ^ [P5: a > $o,R4: a > set_Pr1410444956374649615_a_b_a] :
          ( collec4077122849528628852_a_b_a
          @ ( produc206302895227458626_b_a_o
            @ ( produc5371551483969925184_b_a_o
              @ ^ [X5: a,Y6: mappin2365780136655028071_a_b_a] :
                  ( produc2053338206788791453_b_a_o
                  @ ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] :
                      ( ( X5 = X4 )
                      & ( P5 @ X4 )
                      & ( member1544477001419906904_a_b_a @ ( produc5677138682576874855_a_b_a @ Y6 @ Y4 ) @ ( R4 @ X4 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_381_same__fst__def,axiom,
    ( same_fst_a_b
    = ( ^ [P5: a > $o,R4: a > set_Product_prod_b_b] :
          ( collec7968472090841016338od_a_b
          @ ( produc5413562937804622316_a_b_o
            @ ( produc8890515534965058434_a_b_o
              @ ^ [X5: a,Y6: b] :
                  ( produc3537405659489547051_a_b_o
                  @ ^ [X4: a,Y4: b] :
                      ( ( X5 = X4 )
                      & ( P5 @ X4 )
                      & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y6 @ Y4 ) @ ( R4 @ X4 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_382_mmap__fold_H__def,axiom,
    ( mmap_f6064999285056190608_a_b_a
    = ( ^ [M3: list_P6684558232516339260_d_nat,E5: mappin2365780136655028071_a_b_a,F2: produc6343861673719254644_a_b_a > produc6343861673719254644_a_b_a,C3: produc5456268773499131775_d_nat > produc5456268773499131775_d_nat > produc5456268773499131775_d_nat,R2: list_P6684558232516339260_d_nat] :
          ( foldl_3722756957450160079_d_nat
          @ ( produc9028848012815890814_a_b_a
            @ ^ [S2: list_P6684558232516339260_d_nat,G2: mappin2365780136655028071_a_b_a,P6: produc1990351791824763820_d_nat] :
                ( produc4732899263663970435_a_b_a
                @ ^ [A5: produc1990351791824763820_d_nat,E6: mappin2365780136655028071_a_b_a] :
                    ( produc4991078678549245937_a_b_a
                    @ ^ [K: a,V: produc5456268773499131775_d_nat] : ( produc2616710729536263484_a_b_a @ ( mmap_c2205071097579735977_d_nat @ K @ V @ C3 @ S2 ) @ E6 )
                    @ A5 )
                @ ( F2 @ ( produc1681437058057156268_a_b_a @ P6 @ G2 ) ) ) )
          @ ( produc2616710729536263484_a_b_a @ R2 @ E5 )
          @ M3 ) ) ) ).

% mmap_fold'_def
thf(fact_383_The__split__eq,axiom,
    ! [X: a,Y5: mappin4996136453398522101_b_a_a] :
      ( ( the_Pr1762997456034221349_b_a_a
        @ ( produc5272303750545776437_a_a_o
          @ ^ [X5: a,Y6: mappin4996136453398522101_b_a_a] :
              ( ( X = X5 )
              & ( Y5 = Y6 ) ) ) )
      = ( produc4259011374121840526_b_a_a @ X @ Y5 ) ) ).

% The_split_eq
thf(fact_384_The__split__eq,axiom,
    ! [X: a,Y5: list_P6684558232516339260_d_nat] :
      ( ( the_Pr1862402717852443426_d_nat
        @ ( produc968428876690328712_nat_o
          @ ^ [X5: a,Y6: list_P6684558232516339260_d_nat] :
              ( ( X = X5 )
              & ( Y5 = Y6 ) ) ) )
      = ( produc919746265332163299_d_nat @ X @ Y5 ) ) ).

% The_split_eq
thf(fact_385_The__split__eq,axiom,
    ! [X: a,Y5: list_P2992919015626629540_a_b_a] :
      ( ( the_Pr4937942044906663946_a_b_a
        @ ( produc3087838301718062496_b_a_o
          @ ^ [X5: a,Y6: list_P2992919015626629540_a_b_a] :
              ( ( X = X5 )
              & ( Y5 = Y6 ) ) ) )
      = ( produc6994869176599099851_a_b_a @ X @ Y5 ) ) ).

% The_split_eq
thf(fact_386_The__split__eq,axiom,
    ! [X: a,Y5: list_P1467974523212632366od_a_b] :
      ( ( the_Pr66210465194921182od_a_b
        @ ( produc7670699999246354108_a_b_o
          @ ^ [X5: a,Y6: list_P1467974523212632366od_a_b] :
              ( ( X = X5 )
              & ( Y5 = Y6 ) ) ) )
      = ( produc6643264353547226055od_a_b @ X @ Y5 ) ) ).

% The_split_eq
thf(fact_387_The__split__eq,axiom,
    ! [X: a,Y5: a] :
      ( ( the_Product_prod_a_a
        @ ( produc6436628058953941356_a_a_o
          @ ^ [X5: a,Y6: a] :
              ( ( X = X5 )
              & ( Y5 = Y6 ) ) ) )
      = ( product_Pair_a_a @ X @ Y5 ) ) ).

% The_split_eq
thf(fact_388_The__split__eq,axiom,
    ! [X: a,Y5: produc5456268773499131775_d_nat] :
      ( ( the_Pr6008708066028318565_d_nat
        @ ( produc1905287380190322693_nat_o
          @ ^ [X5: a,Y6: produc5456268773499131775_d_nat] :
              ( ( X = X5 )
              & ( Y5 = Y6 ) ) ) )
      = ( produc5865182755557626534_d_nat @ X @ Y5 ) ) ).

% The_split_eq
thf(fact_389_The__split__eq,axiom,
    ! [X: a,Y5: b] :
      ( ( the_Product_prod_a_b
        @ ( produc3537405659489547051_a_b_o
          @ ^ [X5: a,Y6: b] :
              ( ( X = X5 )
              & ( Y5 = Y6 ) ) ) )
      = ( product_Pair_a_b @ X @ Y5 ) ) ).

% The_split_eq
thf(fact_390_The__split__eq,axiom,
    ! [X: produc1990351791824763820_d_nat,Y5: mappin2365780136655028071_a_b_a] :
      ( ( the_Pr6227858779215488635_a_b_a
        @ ( produc6407700812349938327_b_a_o
          @ ^ [X5: produc1990351791824763820_d_nat,Y6: mappin2365780136655028071_a_b_a] :
              ( ( X = X5 )
              & ( Y5 = Y6 ) ) ) )
      = ( produc1681437058057156268_a_b_a @ X @ Y5 ) ) ).

% The_split_eq
thf(fact_391_The__split__eq,axiom,
    ! [X: list_P6684558232516339260_d_nat,Y5: mappin2365780136655028071_a_b_a] :
      ( ( the_Pr3099714871038386443_a_b_a
        @ ( produc6164275942970966535_b_a_o
          @ ^ [X5: list_P6684558232516339260_d_nat,Y6: mappin2365780136655028071_a_b_a] :
              ( ( X = X5 )
              & ( Y5 = Y6 ) ) ) )
      = ( produc2616710729536263484_a_b_a @ X @ Y5 ) ) ).

% The_split_eq
thf(fact_392_The__split__eq,axiom,
    ! [X: a,Y5: mappin2365780136655028071_a_b_a] :
      ( ( the_Pr8288891123593341133_a_b_a
        @ ( produc2053338206788791453_b_a_o
          @ ^ [X5: a,Y6: mappin2365780136655028071_a_b_a] :
              ( ( X = X5 )
              & ( Y5 = Y6 ) ) ) )
      = ( produc1583556749610718734_a_b_a @ X @ Y5 ) ) ).

% The_split_eq
thf(fact_393_ssubst__Pair__rhs,axiom,
    ! [R3: a,S3: mappin4996136453398522101_b_a_a,R: set_Pr7659163277375235966_b_a_a,S4: mappin4996136453398522101_b_a_a] :
      ( ( member7689197028375331655_b_a_a @ ( produc4259011374121840526_b_a_a @ R3 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member7689197028375331655_b_a_a @ ( produc4259011374121840526_b_a_a @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_394_ssubst__Pair__rhs,axiom,
    ! [R3: a,S3: list_P6684558232516339260_d_nat,R: set_Pr8515993315433149983_d_nat,S4: list_P6684558232516339260_d_nat] :
      ( ( member8047447634869378944_d_nat @ ( produc919746265332163299_d_nat @ R3 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member8047447634869378944_d_nat @ ( produc919746265332163299_d_nat @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_395_ssubst__Pair__rhs,axiom,
    ! [R3: a,S3: list_P2992919015626629540_a_b_a,R: set_Pr5765251437205137415_a_b_a,S4: list_P2992919015626629540_a_b_a] :
      ( ( member378967531903691624_a_b_a @ ( produc6994869176599099851_a_b_a @ R3 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member378967531903691624_a_b_a @ ( produc6994869176599099851_a_b_a @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_396_ssubst__Pair__rhs,axiom,
    ! [R3: a,S3: list_P1467974523212632366od_a_b,R: set_Pr7142564897319301815od_a_b,S4: list_P1467974523212632366od_a_b] :
      ( ( member8745141763671436032od_a_b @ ( produc6643264353547226055od_a_b @ R3 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member8745141763671436032od_a_b @ ( produc6643264353547226055od_a_b @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_397_ssubst__Pair__rhs,axiom,
    ! [R3: a,S3: a,R: set_Product_prod_a_a,S4: a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R3 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_398_ssubst__Pair__rhs,axiom,
    ! [R3: a,S3: produc5456268773499131775_d_nat,R: set_Pr8583615357880026082_d_nat,S4: produc5456268773499131775_d_nat] :
      ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ R3 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_399_ssubst__Pair__rhs,axiom,
    ! [R3: a,S3: b,R: set_Product_prod_a_b,S4: b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ R3 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_400_ssubst__Pair__rhs,axiom,
    ! [R3: produc1990351791824763820_d_nat,S3: mappin2365780136655028071_a_b_a,R: set_Pr8226792695654363348_a_b_a,S4: mappin2365780136655028071_a_b_a] :
      ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ R3 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_401_ssubst__Pair__rhs,axiom,
    ! [R3: list_P6684558232516339260_d_nat,S3: mappin2365780136655028071_a_b_a,R: set_Pr2043227878544749668_a_b_a,S4: mappin2365780136655028071_a_b_a] :
      ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ R3 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_402_ssubst__Pair__rhs,axiom,
    ! [R3: a,S3: mappin2365780136655028071_a_b_a,R: set_Pr2084289749507318346_a_b_a,S4: mappin2365780136655028071_a_b_a] :
      ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ R3 @ S3 ) @ R )
     => ( ( S4 = S3 )
       => ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ R3 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_403_in__inv__image,axiom,
    ! [X: a,Y5: a,R3: set_Product_prod_a_a,F: a > a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y5 ) @ ( inv_image_a_a @ R3 @ F ) )
      = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ X ) @ ( F @ Y5 ) ) @ R3 ) ) ).

% in_inv_image
thf(fact_404_lookup__update__neq,axiom,
    ! [K2: produc7822699449245016212_a_b_a,K3: produc7822699449245016212_a_b_a,V2: a,M: mappin4996136453398522101_b_a_a] :
      ( ( K2 != K3 )
     => ( ( lookup1409243802988840364_b_a_a @ ( update5685058447907601597_b_a_a @ K2 @ V2 @ M ) @ K3 )
        = ( lookup1409243802988840364_b_a_a @ M @ K3 ) ) ) ).

% lookup_update_neq
thf(fact_405_lookup__update__neq,axiom,
    ! [K2: list_P6684558232516339260_d_nat,K3: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( K2 != K3 )
     => ( ( lookup4661578797684577249_a_b_a @ ( update5108779570945132176_a_b_a @ K2 @ V2 @ M ) @ K3 )
        = ( lookup4661578797684577249_a_b_a @ M @ K3 ) ) ) ).

% lookup_update_neq
thf(fact_406_lookup__update__neq,axiom,
    ! [K2: a,K3: a,V2: produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( K2 != K3 )
     => ( ( lookup8480750236328838337_d_nat @ ( update6116250764560905298_d_nat @ K2 @ V2 @ M ) @ K3 )
        = ( lookup8480750236328838337_d_nat @ M @ K3 ) ) ) ).

% lookup_update_neq
thf(fact_407_lookup__update__neq,axiom,
    ! [K2: a,K3: a,V2: mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( K2 != K3 )
     => ( ( lookup1261341372242361769_a_b_a @ ( update5632971445410033082_a_b_a @ K2 @ V2 @ M ) @ K3 )
        = ( lookup1261341372242361769_a_b_a @ M @ K3 ) ) ) ).

% lookup_update_neq
thf(fact_408_lookup__update__neq,axiom,
    ! [K2: a,K3: a,V2: b,M: mapping_a_b] :
      ( ( K2 != K3 )
     => ( ( lookup_a_b @ ( update_a_b @ K2 @ V2 @ M ) @ K3 )
        = ( lookup_a_b @ M @ K3 ) ) ) ).

% lookup_update_neq
thf(fact_409_lookup__update__neq,axiom,
    ! [K2: product_prod_a_b,K3: product_prod_a_b,V2: a,M: mappin2365780136655028071_a_b_a] :
      ( ( K2 != K3 )
     => ( ( lookup863726980359885586_a_b_a @ ( update751575294834384705_a_b_a @ K2 @ V2 @ M ) @ K3 )
        = ( lookup863726980359885586_a_b_a @ M @ K3 ) ) ) ).

% lookup_update_neq
thf(fact_410_same__fstI,axiom,
    ! [P: a > $o,X: a,Y7: mappin4996136453398522101_b_a_a,Y5: mappin4996136453398522101_b_a_a,R: a > set_Pr8669092719848699463_b_a_a] :
      ( ( P @ X )
     => ( ( member9145324461869381776_b_a_a @ ( produc4990582932295648215_b_a_a @ Y7 @ Y5 ) @ ( R @ X ) )
       => ( member7358394357974379728_b_a_a @ ( produc8689759926502962327_b_a_a @ ( produc4259011374121840526_b_a_a @ X @ Y7 ) @ ( produc4259011374121840526_b_a_a @ X @ Y5 ) ) @ ( same_f8131973087259725539_b_a_a @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_411_same__fstI,axiom,
    ! [P: a > $o,X: a,Y7: list_P6684558232516339260_d_nat,Y5: list_P6684558232516339260_d_nat,R: a > set_Pr9022597740479335481_d_nat] :
      ( ( P @ X )
     => ( ( member2190520509249938818_d_nat @ ( produc3155371298979016209_d_nat @ Y7 @ Y5 ) @ ( R @ X ) )
       => ( member6594491860770145628_d_nat @ ( produc3059042386900993131_d_nat @ ( produc919746265332163299_d_nat @ X @ Y7 ) @ ( produc919746265332163299_d_nat @ X @ Y5 ) ) @ ( same_f9108216109512909262_d_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_412_same__fstI,axiom,
    ! [P: a > $o,X: a,Y7: list_P2992919015626629540_a_b_a,Y5: list_P2992919015626629540_a_b_a,R: a > set_Pr295230314114413961_a_b_a] :
      ( ( P @ X )
     => ( ( member6649184025540717266_a_b_a @ ( produc1725085198979854177_a_b_a @ Y7 @ Y5 ) @ ( R @ X ) )
       => ( member5568475963910572844_a_b_a @ ( produc6901241635017770555_a_b_a @ ( produc6994869176599099851_a_b_a @ X @ Y7 ) @ ( produc6994869176599099851_a_b_a @ X @ Y5 ) ) @ ( same_f2663892498176857654_a_b_a @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_413_same__fstI,axiom,
    ! [P: a > $o,X: a,Y7: list_P1467974523212632366od_a_b,Y5: list_P1467974523212632366od_a_b,R: a > set_Pr3015088788472578951od_a_b] :
      ( ( P @ X )
     => ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Y7 @ Y5 ) @ ( R @ X ) )
       => ( member2055843342231799568od_a_b @ ( produc5146089164249824087od_a_b @ ( produc6643264353547226055od_a_b @ X @ Y7 ) @ ( produc6643264353547226055od_a_b @ X @ Y5 ) ) @ ( same_f1164340394720710428od_a_b @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_414_same__fstI,axiom,
    ! [P: a > $o,X: a,Y7: a,Y5: a,R: a > set_Product_prod_a_a] :
      ( ( P @ X )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y7 @ Y5 ) @ ( R @ X ) )
       => ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( product_Pair_a_a @ X @ Y7 ) @ ( product_Pair_a_a @ X @ Y5 ) ) @ ( same_fst_a_a @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_415_same__fstI,axiom,
    ! [P: a > $o,X: a,Y7: produc5456268773499131775_d_nat,Y5: produc5456268773499131775_d_nat,R: a > set_Pr2853997496678843199_d_nat] :
      ( ( P @ X )
     => ( ( member4240823983429516168_d_nat @ ( produc3096269196918525335_d_nat @ Y7 @ Y5 ) @ ( R @ X ) )
       => ( member5807866841158083298_d_nat @ ( produc3551437613648685937_d_nat @ ( produc5865182755557626534_d_nat @ X @ Y7 ) @ ( produc5865182755557626534_d_nat @ X @ Y5 ) ) @ ( same_f3903536699954488849_d_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_416_same__fstI,axiom,
    ! [P: a > $o,X: a,Y7: b,Y5: b,R: a > set_Product_prod_b_b] :
      ( ( P @ X )
     => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ Y7 @ Y5 ) @ ( R @ X ) )
       => ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ ( product_Pair_a_b @ X @ Y7 ) @ ( product_Pair_a_b @ X @ Y5 ) ) @ ( same_fst_a_b @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_417_same__fstI,axiom,
    ! [P: produc1990351791824763820_d_nat > $o,X: produc1990351791824763820_d_nat,Y7: mappin2365780136655028071_a_b_a,Y5: mappin2365780136655028071_a_b_a,R: produc1990351791824763820_d_nat > set_Pr1410444956374649615_a_b_a] :
      ( ( P @ X )
     => ( ( member1544477001419906904_a_b_a @ ( produc5677138682576874855_a_b_a @ Y7 @ Y5 ) @ ( R @ X ) )
       => ( member6784204726113340880_a_b_a @ ( produc873130865258362007_a_b_a @ ( produc1681437058057156268_a_b_a @ X @ Y7 ) @ ( produc1681437058057156268_a_b_a @ X @ Y5 ) ) @ ( same_f8862508556523255425_a_b_a @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_418_same__fstI,axiom,
    ! [P: list_P6684558232516339260_d_nat > $o,X: list_P6684558232516339260_d_nat,Y7: mappin2365780136655028071_a_b_a,Y5: mappin2365780136655028071_a_b_a,R: list_P6684558232516339260_d_nat > set_Pr1410444956374649615_a_b_a] :
      ( ( P @ X )
     => ( ( member1544477001419906904_a_b_a @ ( produc5677138682576874855_a_b_a @ Y7 @ Y5 ) @ ( R @ X ) )
       => ( member6904694298881649744_a_b_a @ ( produc8988583186931504919_a_b_a @ ( produc2616710729536263484_a_b_a @ X @ Y7 ) @ ( produc2616710729536263484_a_b_a @ X @ Y5 ) ) @ ( same_f2554954299287463569_a_b_a @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_419_same__fstI,axiom,
    ! [P: a > $o,X: a,Y7: mappin2365780136655028071_a_b_a,Y5: mappin2365780136655028071_a_b_a,R: a > set_Pr1410444956374649615_a_b_a] :
      ( ( P @ X )
     => ( ( member1544477001419906904_a_b_a @ ( produc5677138682576874855_a_b_a @ Y7 @ Y5 ) @ ( R @ X ) )
       => ( member8877889123357902642_a_b_a @ ( produc9025683278983384001_a_b_a @ ( produc1583556749610718734_a_b_a @ X @ Y7 ) @ ( produc1583556749610718734_a_b_a @ X @ Y5 ) ) @ ( same_f5330524035721250553_a_b_a @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_420_lookup__default__update__neq,axiom,
    ! [K2: list_P6684558232516339260_d_nat,K3: list_P6684558232516339260_d_nat,D2: mappin2365780136655028071_a_b_a,V2: mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( K2 != K3 )
     => ( ( lookup1777753518785040938_d_nat @ D2 @ ( update5108779570945132176_a_b_a @ K2 @ V2 @ M ) @ K3 )
        = ( lookup1777753518785040938_d_nat @ D2 @ M @ K3 ) ) ) ).

% lookup_default_update_neq
thf(fact_421_lookup__default__update__neq,axiom,
    ! [K2: a,K3: a,D2: produc5456268773499131775_d_nat,V2: produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( K2 != K3 )
     => ( ( lookup2689911013986210168_nat_a @ D2 @ ( update6116250764560905298_d_nat @ K2 @ V2 @ M ) @ K3 )
        = ( lookup2689911013986210168_nat_a @ D2 @ M @ K3 ) ) ) ).

% lookup_default_update_neq
thf(fact_422_lookup__default__update__neq,axiom,
    ! [K2: a,K3: a,D2: mappin2365780136655028071_a_b_a,V2: mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( K2 != K3 )
     => ( ( lookup4260590398720651408_b_a_a @ D2 @ ( update5632971445410033082_a_b_a @ K2 @ V2 @ M ) @ K3 )
        = ( lookup4260590398720651408_b_a_a @ D2 @ M @ K3 ) ) ) ).

% lookup_default_update_neq
thf(fact_423_lookup__default__update__neq,axiom,
    ! [K2: a,K3: a,D2: b,V2: b,M: mapping_a_b] :
      ( ( K2 != K3 )
     => ( ( lookup_default_b_a @ D2 @ ( update_a_b @ K2 @ V2 @ M ) @ K3 )
        = ( lookup_default_b_a @ D2 @ M @ K3 ) ) ) ).

% lookup_default_update_neq
thf(fact_424_lookup__default__update__neq,axiom,
    ! [K2: product_prod_a_b,K3: product_prod_a_b,D2: a,V2: a,M: mappin2365780136655028071_a_b_a] :
      ( ( K2 != K3 )
     => ( ( lookup592102128023449775od_a_b @ D2 @ ( update751575294834384705_a_b_a @ K2 @ V2 @ M ) @ K3 )
        = ( lookup592102128023449775od_a_b @ D2 @ M @ K3 ) ) ) ).

% lookup_default_update_neq
thf(fact_425_lookup__default__update_H,axiom,
    ! [K2: list_P6684558232516339260_d_nat,K3: list_P6684558232516339260_d_nat,D2: mappin2365780136655028071_a_b_a,V2: mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( ( K2 = K3 )
       => ( ( lookup1777753518785040938_d_nat @ D2 @ ( update5108779570945132176_a_b_a @ K2 @ V2 @ M ) @ K3 )
          = V2 ) )
      & ( ( K2 != K3 )
       => ( ( lookup1777753518785040938_d_nat @ D2 @ ( update5108779570945132176_a_b_a @ K2 @ V2 @ M ) @ K3 )
          = ( lookup1777753518785040938_d_nat @ D2 @ M @ K3 ) ) ) ) ).

% lookup_default_update'
thf(fact_426_lookup__default__update_H,axiom,
    ! [K2: a,K3: a,D2: produc5456268773499131775_d_nat,V2: produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( ( K2 = K3 )
       => ( ( lookup2689911013986210168_nat_a @ D2 @ ( update6116250764560905298_d_nat @ K2 @ V2 @ M ) @ K3 )
          = V2 ) )
      & ( ( K2 != K3 )
       => ( ( lookup2689911013986210168_nat_a @ D2 @ ( update6116250764560905298_d_nat @ K2 @ V2 @ M ) @ K3 )
          = ( lookup2689911013986210168_nat_a @ D2 @ M @ K3 ) ) ) ) ).

% lookup_default_update'
thf(fact_427_lookup__default__update_H,axiom,
    ! [K2: a,K3: a,D2: mappin2365780136655028071_a_b_a,V2: mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( ( K2 = K3 )
       => ( ( lookup4260590398720651408_b_a_a @ D2 @ ( update5632971445410033082_a_b_a @ K2 @ V2 @ M ) @ K3 )
          = V2 ) )
      & ( ( K2 != K3 )
       => ( ( lookup4260590398720651408_b_a_a @ D2 @ ( update5632971445410033082_a_b_a @ K2 @ V2 @ M ) @ K3 )
          = ( lookup4260590398720651408_b_a_a @ D2 @ M @ K3 ) ) ) ) ).

% lookup_default_update'
thf(fact_428_lookup__default__update_H,axiom,
    ! [K2: a,K3: a,D2: b,V2: b,M: mapping_a_b] :
      ( ( ( K2 = K3 )
       => ( ( lookup_default_b_a @ D2 @ ( update_a_b @ K2 @ V2 @ M ) @ K3 )
          = V2 ) )
      & ( ( K2 != K3 )
       => ( ( lookup_default_b_a @ D2 @ ( update_a_b @ K2 @ V2 @ M ) @ K3 )
          = ( lookup_default_b_a @ D2 @ M @ K3 ) ) ) ) ).

% lookup_default_update'
thf(fact_429_lookup__default__update_H,axiom,
    ! [K2: product_prod_a_b,K3: product_prod_a_b,D2: a,V2: a,M: mappin2365780136655028071_a_b_a] :
      ( ( ( K2 = K3 )
       => ( ( lookup592102128023449775od_a_b @ D2 @ ( update751575294834384705_a_b_a @ K2 @ V2 @ M ) @ K3 )
          = V2 ) )
      & ( ( K2 != K3 )
       => ( ( lookup592102128023449775od_a_b @ D2 @ ( update751575294834384705_a_b_a @ K2 @ V2 @ M ) @ K3 )
          = ( lookup592102128023449775od_a_b @ D2 @ M @ K3 ) ) ) ) ).

% lookup_default_update'
thf(fact_430_All__mapping__update__iff,axiom,
    ! [K2: product_prod_a_b,V2: a,M: mappin2365780136655028071_a_b_a,P: product_prod_a_b > a > $o] :
      ( ( all_ma2923619520195476869_a_b_a @ ( update751575294834384705_a_b_a @ K2 @ V2 @ M ) @ P )
      = ( ( P @ K2 @ V2 )
        & ( all_ma2923619520195476869_a_b_a @ M
          @ ^ [K4: product_prod_a_b,V3: a] :
              ( ( K2 = K4 )
              | ( P @ K4 @ V3 ) ) ) ) ) ).

% All_mapping_update_iff
thf(fact_431_All__mapping__update__iff,axiom,
    ! [K2: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a,P: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > $o] :
      ( ( all_ma3172627230583229140_a_b_a @ ( update5108779570945132176_a_b_a @ K2 @ V2 @ M ) @ P )
      = ( ( P @ K2 @ V2 )
        & ( all_ma3172627230583229140_a_b_a @ M
          @ ^ [K4: list_P6684558232516339260_d_nat,V3: mappin2365780136655028071_a_b_a] :
              ( ( K2 = K4 )
              | ( P @ K4 @ V3 ) ) ) ) ) ).

% All_mapping_update_iff
thf(fact_432_All__mapping__update__iff,axiom,
    ! [K2: a,V2: produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat,P: a > produc5456268773499131775_d_nat > $o] :
      ( ( all_ma1625059968680912654_d_nat @ ( update6116250764560905298_d_nat @ K2 @ V2 @ M ) @ P )
      = ( ( P @ K2 @ V2 )
        & ( all_ma1625059968680912654_d_nat @ M
          @ ^ [K4: a,V3: produc5456268773499131775_d_nat] :
              ( ( K2 = K4 )
              | ( P @ K4 @ V3 ) ) ) ) ) ).

% All_mapping_update_iff
thf(fact_433_All__mapping__update__iff,axiom,
    ! [K2: a,V2: mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a,P: a > mappin2365780136655028071_a_b_a > $o] :
      ( ( all_ma2913212136517639798_a_b_a @ ( update5632971445410033082_a_b_a @ K2 @ V2 @ M ) @ P )
      = ( ( P @ K2 @ V2 )
        & ( all_ma2913212136517639798_a_b_a @ M
          @ ^ [K4: a,V3: mappin2365780136655028071_a_b_a] :
              ( ( K2 = K4 )
              | ( P @ K4 @ V3 ) ) ) ) ) ).

% All_mapping_update_iff
thf(fact_434_All__mapping__update__iff,axiom,
    ! [K2: a,V2: b,M: mapping_a_b,P: a > b > $o] :
      ( ( all_mapping_a_b @ ( update_a_b @ K2 @ V2 @ M ) @ P )
      = ( ( P @ K2 @ V2 )
        & ( all_mapping_a_b @ M
          @ ^ [K4: a,V3: b] :
              ( ( K2 = K4 )
              | ( P @ K4 @ V3 ) ) ) ) ) ).

% All_mapping_update_iff
thf(fact_435_lookup__default__update,axiom,
    ! [D2: mappin2365780136655028071_a_b_a,K2: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( lookup1777753518785040938_d_nat @ D2 @ ( update5108779570945132176_a_b_a @ K2 @ V2 @ M ) @ K2 )
      = V2 ) ).

% lookup_default_update
thf(fact_436_lookup__default__update,axiom,
    ! [D2: produc5456268773499131775_d_nat,K2: a,V2: produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( lookup2689911013986210168_nat_a @ D2 @ ( update6116250764560905298_d_nat @ K2 @ V2 @ M ) @ K2 )
      = V2 ) ).

% lookup_default_update
thf(fact_437_lookup__default__update,axiom,
    ! [D2: mappin2365780136655028071_a_b_a,K2: a,V2: mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( lookup4260590398720651408_b_a_a @ D2 @ ( update5632971445410033082_a_b_a @ K2 @ V2 @ M ) @ K2 )
      = V2 ) ).

% lookup_default_update
thf(fact_438_lookup__default__update,axiom,
    ! [D2: b,K2: a,V2: b,M: mapping_a_b] :
      ( ( lookup_default_b_a @ D2 @ ( update_a_b @ K2 @ V2 @ M ) @ K2 )
      = V2 ) ).

% lookup_default_update
thf(fact_439_lookup__default__update,axiom,
    ! [D2: a,K2: product_prod_a_b,V2: a,M: mappin2365780136655028071_a_b_a] :
      ( ( lookup592102128023449775od_a_b @ D2 @ ( update751575294834384705_a_b_a @ K2 @ V2 @ M ) @ K2 )
      = V2 ) ).

% lookup_default_update
thf(fact_440_All__mapping__update,axiom,
    ! [P: product_prod_a_b > a > $o,K2: product_prod_a_b,V2: a,M: mappin2365780136655028071_a_b_a] :
      ( ( P @ K2 @ V2 )
     => ( ( all_ma2923619520195476869_a_b_a @ M
          @ ^ [K4: product_prod_a_b,V3: a] :
              ( ( K2 = K4 )
              | ( P @ K4 @ V3 ) ) )
       => ( all_ma2923619520195476869_a_b_a @ ( update751575294834384705_a_b_a @ K2 @ V2 @ M ) @ P ) ) ) ).

% All_mapping_update
thf(fact_441_All__mapping__update,axiom,
    ! [P: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > $o,K2: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( P @ K2 @ V2 )
     => ( ( all_ma3172627230583229140_a_b_a @ M
          @ ^ [K4: list_P6684558232516339260_d_nat,V3: mappin2365780136655028071_a_b_a] :
              ( ( K2 = K4 )
              | ( P @ K4 @ V3 ) ) )
       => ( all_ma3172627230583229140_a_b_a @ ( update5108779570945132176_a_b_a @ K2 @ V2 @ M ) @ P ) ) ) ).

% All_mapping_update
thf(fact_442_All__mapping__update,axiom,
    ! [P: a > produc5456268773499131775_d_nat > $o,K2: a,V2: produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( P @ K2 @ V2 )
     => ( ( all_ma1625059968680912654_d_nat @ M
          @ ^ [K4: a,V3: produc5456268773499131775_d_nat] :
              ( ( K2 = K4 )
              | ( P @ K4 @ V3 ) ) )
       => ( all_ma1625059968680912654_d_nat @ ( update6116250764560905298_d_nat @ K2 @ V2 @ M ) @ P ) ) ) ).

% All_mapping_update
thf(fact_443_All__mapping__update,axiom,
    ! [P: a > mappin2365780136655028071_a_b_a > $o,K2: a,V2: mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( P @ K2 @ V2 )
     => ( ( all_ma2913212136517639798_a_b_a @ M
          @ ^ [K4: a,V3: mappin2365780136655028071_a_b_a] :
              ( ( K2 = K4 )
              | ( P @ K4 @ V3 ) ) )
       => ( all_ma2913212136517639798_a_b_a @ ( update5632971445410033082_a_b_a @ K2 @ V2 @ M ) @ P ) ) ) ).

% All_mapping_update
thf(fact_444_All__mapping__update,axiom,
    ! [P: a > b > $o,K2: a,V2: b,M: mapping_a_b] :
      ( ( P @ K2 @ V2 )
     => ( ( all_mapping_a_b @ M
          @ ^ [K4: a,V3: b] :
              ( ( K2 = K4 )
              | ( P @ K4 @ V3 ) ) )
       => ( all_mapping_a_b @ ( update_a_b @ K2 @ V2 @ M ) @ P ) ) ) ).

% All_mapping_update
thf(fact_445_update__update_I1_J,axiom,
    ! [K2: product_prod_a_b,V2: a,W: a,M: mappin2365780136655028071_a_b_a] :
      ( ( update751575294834384705_a_b_a @ K2 @ V2 @ ( update751575294834384705_a_b_a @ K2 @ W @ M ) )
      = ( update751575294834384705_a_b_a @ K2 @ V2 @ M ) ) ).

% update_update(1)
thf(fact_446_update__update_I1_J,axiom,
    ! [K2: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a,W: mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( update5108779570945132176_a_b_a @ K2 @ V2 @ ( update5108779570945132176_a_b_a @ K2 @ W @ M ) )
      = ( update5108779570945132176_a_b_a @ K2 @ V2 @ M ) ) ).

% update_update(1)
thf(fact_447_update__update_I1_J,axiom,
    ! [K2: a,V2: produc5456268773499131775_d_nat,W: produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( update6116250764560905298_d_nat @ K2 @ V2 @ ( update6116250764560905298_d_nat @ K2 @ W @ M ) )
      = ( update6116250764560905298_d_nat @ K2 @ V2 @ M ) ) ).

% update_update(1)
thf(fact_448_update__update_I1_J,axiom,
    ! [K2: a,V2: mappin2365780136655028071_a_b_a,W: mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( update5632971445410033082_a_b_a @ K2 @ V2 @ ( update5632971445410033082_a_b_a @ K2 @ W @ M ) )
      = ( update5632971445410033082_a_b_a @ K2 @ V2 @ M ) ) ).

% update_update(1)
thf(fact_449_update__update_I1_J,axiom,
    ! [K2: a,V2: b,W: b,M: mapping_a_b] :
      ( ( update_a_b @ K2 @ V2 @ ( update_a_b @ K2 @ W @ M ) )
      = ( update_a_b @ K2 @ V2 @ M ) ) ).

% update_update(1)
thf(fact_450_update__update_I2_J,axiom,
    ! [K2: product_prod_a_b,L2: product_prod_a_b,V2: a,W: a,M: mappin2365780136655028071_a_b_a] :
      ( ( K2 != L2 )
     => ( ( update751575294834384705_a_b_a @ K2 @ V2 @ ( update751575294834384705_a_b_a @ L2 @ W @ M ) )
        = ( update751575294834384705_a_b_a @ L2 @ W @ ( update751575294834384705_a_b_a @ K2 @ V2 @ M ) ) ) ) ).

% update_update(2)
thf(fact_451_update__update_I2_J,axiom,
    ! [K2: list_P6684558232516339260_d_nat,L2: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a,W: mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( K2 != L2 )
     => ( ( update5108779570945132176_a_b_a @ K2 @ V2 @ ( update5108779570945132176_a_b_a @ L2 @ W @ M ) )
        = ( update5108779570945132176_a_b_a @ L2 @ W @ ( update5108779570945132176_a_b_a @ K2 @ V2 @ M ) ) ) ) ).

% update_update(2)
thf(fact_452_update__update_I2_J,axiom,
    ! [K2: a,L2: a,V2: produc5456268773499131775_d_nat,W: produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( K2 != L2 )
     => ( ( update6116250764560905298_d_nat @ K2 @ V2 @ ( update6116250764560905298_d_nat @ L2 @ W @ M ) )
        = ( update6116250764560905298_d_nat @ L2 @ W @ ( update6116250764560905298_d_nat @ K2 @ V2 @ M ) ) ) ) ).

% update_update(2)
thf(fact_453_update__update_I2_J,axiom,
    ! [K2: a,L2: a,V2: mappin2365780136655028071_a_b_a,W: mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( K2 != L2 )
     => ( ( update5632971445410033082_a_b_a @ K2 @ V2 @ ( update5632971445410033082_a_b_a @ L2 @ W @ M ) )
        = ( update5632971445410033082_a_b_a @ L2 @ W @ ( update5632971445410033082_a_b_a @ K2 @ V2 @ M ) ) ) ) ).

% update_update(2)
thf(fact_454_update__update_I2_J,axiom,
    ! [K2: a,L2: a,V2: b,W: b,M: mapping_a_b] :
      ( ( K2 != L2 )
     => ( ( update_a_b @ K2 @ V2 @ ( update_a_b @ L2 @ W @ M ) )
        = ( update_a_b @ L2 @ W @ ( update_a_b @ K2 @ V2 @ M ) ) ) ) ).

% update_update(2)
thf(fact_455_theI__unique,axiom,
    ! [P: produc7822699449245016212_a_b_a > $o,X: produc7822699449245016212_a_b_a] :
      ( ? [X6: produc7822699449245016212_a_b_a] :
          ( ( P @ X6 )
          & ! [Y3: produc7822699449245016212_a_b_a] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ( P @ X )
        = ( X
          = ( the_Pr8288891123593341133_a_b_a @ P ) ) ) ) ).

% theI_unique
thf(fact_456_theI__unique,axiom,
    ! [P: produc2062766750192416900_a_b_a > $o,X: produc2062766750192416900_a_b_a] :
      ( ? [X6: produc2062766750192416900_a_b_a] :
          ( ( P @ X6 )
          & ! [Y3: produc2062766750192416900_a_b_a] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ( P @ X )
        = ( X
          = ( the_Pr3099714871038386443_a_b_a @ P ) ) ) ) ).

% theI_unique
thf(fact_457_theI__unique,axiom,
    ! [P: produc6343861673719254644_a_b_a > $o,X: produc6343861673719254644_a_b_a] :
      ( ? [X6: produc6343861673719254644_a_b_a] :
          ( ( P @ X6 )
          & ! [Y3: produc6343861673719254644_a_b_a] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ( P @ X )
        = ( X
          = ( the_Pr6227858779215488635_a_b_a @ P ) ) ) ) ).

% theI_unique
thf(fact_458_theI__unique,axiom,
    ! [P: product_prod_a_b > $o,X: product_prod_a_b] :
      ( ? [X6: product_prod_a_b] :
          ( ( P @ X6 )
          & ! [Y3: product_prod_a_b] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ( P @ X )
        = ( X
          = ( the_Product_prod_a_b @ P ) ) ) ) ).

% theI_unique
thf(fact_459_theI__unique,axiom,
    ! [P: produc1990351791824763820_d_nat > $o,X: produc1990351791824763820_d_nat] :
      ( ? [X6: produc1990351791824763820_d_nat] :
          ( ( P @ X6 )
          & ! [Y3: produc1990351791824763820_d_nat] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ( P @ X )
        = ( X
          = ( the_Pr6008708066028318565_d_nat @ P ) ) ) ) ).

% theI_unique
thf(fact_460_the__equality,axiom,
    ! [P: produc7822699449245016212_a_b_a > $o,A: produc7822699449245016212_a_b_a] :
      ( ( P @ A )
     => ( ! [X3: produc7822699449245016212_a_b_a] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( ( the_Pr8288891123593341133_a_b_a @ P )
          = A ) ) ) ).

% the_equality
thf(fact_461_the__equality,axiom,
    ! [P: produc2062766750192416900_a_b_a > $o,A: produc2062766750192416900_a_b_a] :
      ( ( P @ A )
     => ( ! [X3: produc2062766750192416900_a_b_a] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( ( the_Pr3099714871038386443_a_b_a @ P )
          = A ) ) ) ).

% the_equality
thf(fact_462_the__equality,axiom,
    ! [P: produc6343861673719254644_a_b_a > $o,A: produc6343861673719254644_a_b_a] :
      ( ( P @ A )
     => ( ! [X3: produc6343861673719254644_a_b_a] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( ( the_Pr6227858779215488635_a_b_a @ P )
          = A ) ) ) ).

% the_equality
thf(fact_463_the__equality,axiom,
    ! [P: product_prod_a_b > $o,A: product_prod_a_b] :
      ( ( P @ A )
     => ( ! [X3: product_prod_a_b] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( ( the_Product_prod_a_b @ P )
          = A ) ) ) ).

% the_equality
thf(fact_464_the__equality,axiom,
    ! [P: produc1990351791824763820_d_nat > $o,A: produc1990351791824763820_d_nat] :
      ( ( P @ A )
     => ( ! [X3: produc1990351791824763820_d_nat] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( ( the_Pr6008708066028318565_d_nat @ P )
          = A ) ) ) ).

% the_equality
thf(fact_465_the__eq__trivial,axiom,
    ! [A: produc7822699449245016212_a_b_a] :
      ( ( the_Pr8288891123593341133_a_b_a
        @ ^ [X4: produc7822699449245016212_a_b_a] : ( X4 = A ) )
      = A ) ).

% the_eq_trivial
thf(fact_466_the__eq__trivial,axiom,
    ! [A: produc2062766750192416900_a_b_a] :
      ( ( the_Pr3099714871038386443_a_b_a
        @ ^ [X4: produc2062766750192416900_a_b_a] : ( X4 = A ) )
      = A ) ).

% the_eq_trivial
thf(fact_467_the__eq__trivial,axiom,
    ! [A: produc6343861673719254644_a_b_a] :
      ( ( the_Pr6227858779215488635_a_b_a
        @ ^ [X4: produc6343861673719254644_a_b_a] : ( X4 = A ) )
      = A ) ).

% the_eq_trivial
thf(fact_468_the__eq__trivial,axiom,
    ! [A: product_prod_a_b] :
      ( ( the_Product_prod_a_b
        @ ^ [X4: product_prod_a_b] : ( X4 = A ) )
      = A ) ).

% the_eq_trivial
thf(fact_469_the__eq__trivial,axiom,
    ! [A: produc1990351791824763820_d_nat] :
      ( ( the_Pr6008708066028318565_d_nat
        @ ^ [X4: produc1990351791824763820_d_nat] : ( X4 = A ) )
      = A ) ).

% the_eq_trivial
thf(fact_470_the__sym__eq__trivial,axiom,
    ! [X: produc7822699449245016212_a_b_a] :
      ( ( the_Pr8288891123593341133_a_b_a
        @ ( ^ [Y: produc7822699449245016212_a_b_a,Z: produc7822699449245016212_a_b_a] : ( Y = Z )
          @ X ) )
      = X ) ).

% the_sym_eq_trivial
thf(fact_471_the__sym__eq__trivial,axiom,
    ! [X: produc2062766750192416900_a_b_a] :
      ( ( the_Pr3099714871038386443_a_b_a
        @ ( ^ [Y: produc2062766750192416900_a_b_a,Z: produc2062766750192416900_a_b_a] : ( Y = Z )
          @ X ) )
      = X ) ).

% the_sym_eq_trivial
thf(fact_472_the__sym__eq__trivial,axiom,
    ! [X: produc6343861673719254644_a_b_a] :
      ( ( the_Pr6227858779215488635_a_b_a
        @ ( ^ [Y: produc6343861673719254644_a_b_a,Z: produc6343861673719254644_a_b_a] : ( Y = Z )
          @ X ) )
      = X ) ).

% the_sym_eq_trivial
thf(fact_473_the__sym__eq__trivial,axiom,
    ! [X: product_prod_a_b] :
      ( ( the_Product_prod_a_b
        @ ( ^ [Y: product_prod_a_b,Z: product_prod_a_b] : ( Y = Z )
          @ X ) )
      = X ) ).

% the_sym_eq_trivial
thf(fact_474_the__sym__eq__trivial,axiom,
    ! [X: produc1990351791824763820_d_nat] :
      ( ( the_Pr6008708066028318565_d_nat
        @ ( ^ [Y: produc1990351791824763820_d_nat,Z: produc1990351791824763820_d_nat] : ( Y = Z )
          @ X ) )
      = X ) ).

% the_sym_eq_trivial
thf(fact_475_foldl__Nil,axiom,
    ! [F: produc2062766750192416900_a_b_a > produc1990351791824763820_d_nat > produc2062766750192416900_a_b_a,A: produc2062766750192416900_a_b_a] :
      ( ( foldl_3722756957450160079_d_nat @ F @ A @ nil_Pr2300798825815202086_d_nat )
      = A ) ).

% foldl_Nil
thf(fact_476_foldl__Nil,axiom,
    ! [F: list_P6684558232516339260_d_nat > produc1990351791824763820_d_nat > list_P6684558232516339260_d_nat,A: list_P6684558232516339260_d_nat] :
      ( ( foldl_5754872400397474315_d_nat @ F @ A @ nil_Pr2300798825815202086_d_nat )
      = A ) ).

% foldl_Nil
thf(fact_477_lex__prod__def,axiom,
    ( lex_prod_a_a
    = ( ^ [Ra: set_Product_prod_a_a,Rb: set_Product_prod_a_a] :
          ( collec10116633892588882od_a_a
          @ ( produc4261069933152462124_a_a_o
            @ ( produc5089340801457792962_a_a_o
              @ ^ [A5: a,B4: a] :
                  ( produc6436628058953941356_a_a_o
                  @ ^ [A6: a,B5: a] :
                      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A5 @ A6 ) @ Ra )
                      | ( ( A5 = A6 )
                        & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B4 @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_478_lex__prod__def,axiom,
    ( lex_pr7112548308683715924_a_b_a
    = ( ^ [Ra: set_Pr9022597740479335481_d_nat,Rb: set_Pr1410444956374649615_a_b_a] :
          ( collec7317118955271278738_a_b_a
          @ ( produc8109344284017171820_b_a_o
            @ ( produc7322125808614349514_b_a_o
              @ ^ [A5: list_P6684558232516339260_d_nat,B4: mappin2365780136655028071_a_b_a] :
                  ( produc6164275942970966535_b_a_o
                  @ ^ [A6: list_P6684558232516339260_d_nat,B5: mappin2365780136655028071_a_b_a] :
                      ( ( member2190520509249938818_d_nat @ ( produc3155371298979016209_d_nat @ A5 @ A6 ) @ Ra )
                      | ( ( A5 = A6 )
                        & ( member1544477001419906904_a_b_a @ ( produc5677138682576874855_a_b_a @ B4 @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_479_lex__prod__def,axiom,
    ( lex_pr6843718498755887812_a_b_a
    = ( ^ [Ra: set_Pr8180828926698915225_d_nat,Rb: set_Pr1410444956374649615_a_b_a] :
          ( collec1164995411054953490_a_b_a
          @ ( produc5089158207452318700_b_a_o
            @ ( produc708482854371332170_b_a_o
              @ ^ [A5: produc1990351791824763820_d_nat,B4: mappin2365780136655028071_a_b_a] :
                  ( produc6407700812349938327_b_a_o
                  @ ^ [A6: produc1990351791824763820_d_nat,B5: mappin2365780136655028071_a_b_a] :
                      ( ( member5807866841158083298_d_nat @ ( produc3551437613648685937_d_nat @ A5 @ A6 ) @ Ra )
                      | ( ( A5 = A6 )
                        & ( member1544477001419906904_a_b_a @ ( produc5677138682576874855_a_b_a @ B4 @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_480_lex__prod__def,axiom,
    ( lex_pr9159876284408421518_d_nat
    = ( ^ [Ra: set_Product_prod_a_a,Rb: set_Pr2853997496678843199_d_nat] :
          ( collec2880425841902469156_d_nat
          @ ( produc4563981736279841938_nat_o
            @ ( produc6235630094606560144_nat_o
              @ ^ [A5: a,B4: produc5456268773499131775_d_nat] :
                  ( produc1905287380190322693_nat_o
                  @ ^ [A6: a,B5: produc5456268773499131775_d_nat] :
                      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A5 @ A6 ) @ Ra )
                      | ( ( A5 = A6 )
                        & ( member4240823983429516168_d_nat @ ( produc3096269196918525335_d_nat @ B4 @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_481_lex__prod__def,axiom,
    ( lex_pr7193167406128026614_a_b_a
    = ( ^ [Ra: set_Product_prod_a_a,Rb: set_Pr1410444956374649615_a_b_a] :
          ( collec4077122849528628852_a_b_a
          @ ( produc206302895227458626_b_a_o
            @ ( produc5371551483969925184_b_a_o
              @ ^ [A5: a,B4: mappin2365780136655028071_a_b_a] :
                  ( produc2053338206788791453_b_a_o
                  @ ^ [A6: a,B5: mappin2365780136655028071_a_b_a] :
                      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A5 @ A6 ) @ Ra )
                      | ( ( A5 = A6 )
                        & ( member1544477001419906904_a_b_a @ ( produc5677138682576874855_a_b_a @ B4 @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_482_lex__prod__def,axiom,
    ( lex_prod_a_b
    = ( ^ [Ra: set_Product_prod_a_a,Rb: set_Product_prod_b_b] :
          ( collec7968472090841016338od_a_b
          @ ( produc5413562937804622316_a_b_o
            @ ( produc8890515534965058434_a_b_o
              @ ^ [A5: a,B4: b] :
                  ( produc3537405659489547051_a_b_o
                  @ ^ [A6: a,B5: b] :
                      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A5 @ A6 ) @ Ra )
                      | ( ( A5 = A6 )
                        & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B4 @ B5 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_483_theI,axiom,
    ! [P: produc7822699449245016212_a_b_a > $o,A: produc7822699449245016212_a_b_a] :
      ( ( P @ A )
     => ( ! [X3: produc7822699449245016212_a_b_a] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( P @ ( the_Pr8288891123593341133_a_b_a @ P ) ) ) ) ).

% theI
thf(fact_484_theI,axiom,
    ! [P: produc2062766750192416900_a_b_a > $o,A: produc2062766750192416900_a_b_a] :
      ( ( P @ A )
     => ( ! [X3: produc2062766750192416900_a_b_a] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( P @ ( the_Pr3099714871038386443_a_b_a @ P ) ) ) ) ).

% theI
thf(fact_485_theI,axiom,
    ! [P: produc6343861673719254644_a_b_a > $o,A: produc6343861673719254644_a_b_a] :
      ( ( P @ A )
     => ( ! [X3: produc6343861673719254644_a_b_a] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( P @ ( the_Pr6227858779215488635_a_b_a @ P ) ) ) ) ).

% theI
thf(fact_486_theI,axiom,
    ! [P: product_prod_a_b > $o,A: product_prod_a_b] :
      ( ( P @ A )
     => ( ! [X3: product_prod_a_b] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( P @ ( the_Product_prod_a_b @ P ) ) ) ) ).

% theI
thf(fact_487_theI,axiom,
    ! [P: produc1990351791824763820_d_nat > $o,A: produc1990351791824763820_d_nat] :
      ( ( P @ A )
     => ( ! [X3: produc1990351791824763820_d_nat] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( P @ ( the_Pr6008708066028318565_d_nat @ P ) ) ) ) ).

% theI
thf(fact_488_theI_H,axiom,
    ! [P: produc7822699449245016212_a_b_a > $o] :
      ( ? [X6: produc7822699449245016212_a_b_a] :
          ( ( P @ X6 )
          & ! [Y3: produc7822699449245016212_a_b_a] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( P @ ( the_Pr8288891123593341133_a_b_a @ P ) ) ) ).

% theI'
thf(fact_489_theI_H,axiom,
    ! [P: produc2062766750192416900_a_b_a > $o] :
      ( ? [X6: produc2062766750192416900_a_b_a] :
          ( ( P @ X6 )
          & ! [Y3: produc2062766750192416900_a_b_a] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( P @ ( the_Pr3099714871038386443_a_b_a @ P ) ) ) ).

% theI'
thf(fact_490_theI_H,axiom,
    ! [P: produc6343861673719254644_a_b_a > $o] :
      ( ? [X6: produc6343861673719254644_a_b_a] :
          ( ( P @ X6 )
          & ! [Y3: produc6343861673719254644_a_b_a] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( P @ ( the_Pr6227858779215488635_a_b_a @ P ) ) ) ).

% theI'
thf(fact_491_theI_H,axiom,
    ! [P: product_prod_a_b > $o] :
      ( ? [X6: product_prod_a_b] :
          ( ( P @ X6 )
          & ! [Y3: product_prod_a_b] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( P @ ( the_Product_prod_a_b @ P ) ) ) ).

% theI'
thf(fact_492_theI_H,axiom,
    ! [P: produc1990351791824763820_d_nat > $o] :
      ( ? [X6: produc1990351791824763820_d_nat] :
          ( ( P @ X6 )
          & ! [Y3: produc1990351791824763820_d_nat] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( P @ ( the_Pr6008708066028318565_d_nat @ P ) ) ) ).

% theI'
thf(fact_493_theI2,axiom,
    ! [P: produc7822699449245016212_a_b_a > $o,A: produc7822699449245016212_a_b_a,Q2: produc7822699449245016212_a_b_a > $o] :
      ( ( P @ A )
     => ( ! [X3: produc7822699449245016212_a_b_a] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( ! [X3: produc7822699449245016212_a_b_a] :
              ( ( P @ X3 )
             => ( Q2 @ X3 ) )
         => ( Q2 @ ( the_Pr8288891123593341133_a_b_a @ P ) ) ) ) ) ).

% theI2
thf(fact_494_theI2,axiom,
    ! [P: produc2062766750192416900_a_b_a > $o,A: produc2062766750192416900_a_b_a,Q2: produc2062766750192416900_a_b_a > $o] :
      ( ( P @ A )
     => ( ! [X3: produc2062766750192416900_a_b_a] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( ! [X3: produc2062766750192416900_a_b_a] :
              ( ( P @ X3 )
             => ( Q2 @ X3 ) )
         => ( Q2 @ ( the_Pr3099714871038386443_a_b_a @ P ) ) ) ) ) ).

% theI2
thf(fact_495_theI2,axiom,
    ! [P: produc6343861673719254644_a_b_a > $o,A: produc6343861673719254644_a_b_a,Q2: produc6343861673719254644_a_b_a > $o] :
      ( ( P @ A )
     => ( ! [X3: produc6343861673719254644_a_b_a] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( ! [X3: produc6343861673719254644_a_b_a] :
              ( ( P @ X3 )
             => ( Q2 @ X3 ) )
         => ( Q2 @ ( the_Pr6227858779215488635_a_b_a @ P ) ) ) ) ) ).

% theI2
thf(fact_496_theI2,axiom,
    ! [P: product_prod_a_b > $o,A: product_prod_a_b,Q2: product_prod_a_b > $o] :
      ( ( P @ A )
     => ( ! [X3: product_prod_a_b] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( ! [X3: product_prod_a_b] :
              ( ( P @ X3 )
             => ( Q2 @ X3 ) )
         => ( Q2 @ ( the_Product_prod_a_b @ P ) ) ) ) ) ).

% theI2
thf(fact_497_theI2,axiom,
    ! [P: produc1990351791824763820_d_nat > $o,A: produc1990351791824763820_d_nat,Q2: produc1990351791824763820_d_nat > $o] :
      ( ( P @ A )
     => ( ! [X3: produc1990351791824763820_d_nat] :
            ( ( P @ X3 )
           => ( X3 = A ) )
       => ( ! [X3: produc1990351791824763820_d_nat] :
              ( ( P @ X3 )
             => ( Q2 @ X3 ) )
         => ( Q2 @ ( the_Pr6008708066028318565_d_nat @ P ) ) ) ) ) ).

% theI2
thf(fact_498_in__lex__prod,axiom,
    ! [A: a,B: mappin4996136453398522101_b_a_a,A2: a,B2: mappin4996136453398522101_b_a_a,R3: set_Product_prod_a_a,S3: set_Pr8669092719848699463_b_a_a] :
      ( ( member7358394357974379728_b_a_a @ ( produc8689759926502962327_b_a_a @ ( produc4259011374121840526_b_a_a @ A @ B ) @ ( produc4259011374121840526_b_a_a @ A2 @ B2 ) ) @ ( lex_pr5951504258779389862_b_a_a @ R3 @ S3 ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R3 )
        | ( ( A = A2 )
          & ( member9145324461869381776_b_a_a @ ( produc4990582932295648215_b_a_a @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_499_in__lex__prod,axiom,
    ! [A: a,B: list_P6684558232516339260_d_nat,A2: a,B2: list_P6684558232516339260_d_nat,R3: set_Product_prod_a_a,S3: set_Pr9022597740479335481_d_nat] :
      ( ( member6594491860770145628_d_nat @ ( produc3059042386900993131_d_nat @ ( produc919746265332163299_d_nat @ A @ B ) @ ( produc919746265332163299_d_nat @ A2 @ B2 ) ) @ ( lex_pr2864253577182015691_d_nat @ R3 @ S3 ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R3 )
        | ( ( A = A2 )
          & ( member2190520509249938818_d_nat @ ( produc3155371298979016209_d_nat @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_500_in__lex__prod,axiom,
    ! [A: a,B: list_P2992919015626629540_a_b_a,A2: a,B2: list_P2992919015626629540_a_b_a,R3: set_Product_prod_a_a,S3: set_Pr295230314114413961_a_b_a] :
      ( ( member5568475963910572844_a_b_a @ ( produc6901241635017770555_a_b_a @ ( produc6994869176599099851_a_b_a @ A @ B ) @ ( produc6994869176599099851_a_b_a @ A2 @ B2 ) ) @ ( lex_pr1898994386625379251_a_b_a @ R3 @ S3 ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R3 )
        | ( ( A = A2 )
          & ( member6649184025540717266_a_b_a @ ( produc1725085198979854177_a_b_a @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_501_in__lex__prod,axiom,
    ! [A: a,B: list_P1467974523212632366od_a_b,A2: a,B2: list_P1467974523212632366od_a_b,R3: set_Product_prod_a_a,S3: set_Pr3015088788472578951od_a_b] :
      ( ( member2055843342231799568od_a_b @ ( produc5146089164249824087od_a_b @ ( produc6643264353547226055od_a_b @ A @ B ) @ ( produc6643264353547226055od_a_b @ A2 @ B2 ) ) @ ( lex_pr1713614234531929055od_a_b @ R3 @ S3 ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R3 )
        | ( ( A = A2 )
          & ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_502_in__lex__prod,axiom,
    ! [A: a,B: a,A2: a,B2: a,R3: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
      ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Pair_a_a @ A2 @ B2 ) ) @ ( lex_prod_a_a @ R3 @ S3 ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R3 )
        | ( ( A = A2 )
          & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_503_in__lex__prod,axiom,
    ! [A: a,B: produc5456268773499131775_d_nat,A2: a,B2: produc5456268773499131775_d_nat,R3: set_Product_prod_a_a,S3: set_Pr2853997496678843199_d_nat] :
      ( ( member5807866841158083298_d_nat @ ( produc3551437613648685937_d_nat @ ( produc5865182755557626534_d_nat @ A @ B ) @ ( produc5865182755557626534_d_nat @ A2 @ B2 ) ) @ ( lex_pr9159876284408421518_d_nat @ R3 @ S3 ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R3 )
        | ( ( A = A2 )
          & ( member4240823983429516168_d_nat @ ( produc3096269196918525335_d_nat @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_504_in__lex__prod,axiom,
    ! [A: a,B: b,A2: a,B2: b,R3: set_Product_prod_a_a,S3: set_Product_prod_b_b] :
      ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ ( product_Pair_a_b @ A @ B ) @ ( product_Pair_a_b @ A2 @ B2 ) ) @ ( lex_prod_a_b @ R3 @ S3 ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R3 )
        | ( ( A = A2 )
          & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_505_in__lex__prod,axiom,
    ! [A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a,A2: produc1990351791824763820_d_nat,B2: mappin2365780136655028071_a_b_a,R3: set_Pr8180828926698915225_d_nat,S3: set_Pr1410444956374649615_a_b_a] :
      ( ( member6784204726113340880_a_b_a @ ( produc873130865258362007_a_b_a @ ( produc1681437058057156268_a_b_a @ A @ B ) @ ( produc1681437058057156268_a_b_a @ A2 @ B2 ) ) @ ( lex_pr6843718498755887812_a_b_a @ R3 @ S3 ) )
      = ( ( member5807866841158083298_d_nat @ ( produc3551437613648685937_d_nat @ A @ A2 ) @ R3 )
        | ( ( A = A2 )
          & ( member1544477001419906904_a_b_a @ ( produc5677138682576874855_a_b_a @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_506_in__lex__prod,axiom,
    ! [A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a,A2: list_P6684558232516339260_d_nat,B2: mappin2365780136655028071_a_b_a,R3: set_Pr9022597740479335481_d_nat,S3: set_Pr1410444956374649615_a_b_a] :
      ( ( member6904694298881649744_a_b_a @ ( produc8988583186931504919_a_b_a @ ( produc2616710729536263484_a_b_a @ A @ B ) @ ( produc2616710729536263484_a_b_a @ A2 @ B2 ) ) @ ( lex_pr7112548308683715924_a_b_a @ R3 @ S3 ) )
      = ( ( member2190520509249938818_d_nat @ ( produc3155371298979016209_d_nat @ A @ A2 ) @ R3 )
        | ( ( A = A2 )
          & ( member1544477001419906904_a_b_a @ ( produc5677138682576874855_a_b_a @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_507_in__lex__prod,axiom,
    ! [A: a,B: mappin2365780136655028071_a_b_a,A2: a,B2: mappin2365780136655028071_a_b_a,R3: set_Product_prod_a_a,S3: set_Pr1410444956374649615_a_b_a] :
      ( ( member8877889123357902642_a_b_a @ ( produc9025683278983384001_a_b_a @ ( produc1583556749610718734_a_b_a @ A @ B ) @ ( produc1583556749610718734_a_b_a @ A2 @ B2 ) ) @ ( lex_pr7193167406128026614_a_b_a @ R3 @ S3 ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R3 )
        | ( ( A = A2 )
          & ( member1544477001419906904_a_b_a @ ( produc5677138682576874855_a_b_a @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_508_mmap__fold__def,axiom,
    ( mmap_f6110892234921423665_a_b_a
    = ( ^ [M3: list_P7611574810425128074_a_b_a,F2: produc2062766750192416900_a_b_a > produc2062766750192416900_a_b_a,C3: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,R2: list_P7611574810425128074_a_b_a] :
          ( foldl_227793924032141979_a_b_a
          @ ^ [S2: list_P7611574810425128074_a_b_a,P6: produc2062766750192416900_a_b_a] :
              ( produc2202866172573321337_a_b_a
              @ ^ [K: list_P6684558232516339260_d_nat,V: mappin2365780136655028071_a_b_a] : ( mmap_c8213366964327184121_a_b_a @ K @ V @ C3 @ S2 )
              @ ( F2 @ P6 ) )
          @ R2
          @ M3 ) ) ) ).

% mmap_fold_def
thf(fact_509_mmap__fold__def,axiom,
    ( mmap_f1190519834387993377_a_b_a
    = ( ^ [M3: list_P4793483941206178810_a_b_a,F2: produc6343861673719254644_a_b_a > produc6343861673719254644_a_b_a,C3: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,R2: list_P4793483941206178810_a_b_a] :
          ( foldl_8135302918598886811_a_b_a
          @ ^ [S2: list_P4793483941206178810_a_b_a,P6: produc6343861673719254644_a_b_a] :
              ( produc7463616454677732345_a_b_a
              @ ^ [K: produc1990351791824763820_d_nat,V: mappin2365780136655028071_a_b_a] : ( mmap_c5580072897124408041_a_b_a @ K @ V @ C3 @ S2 )
              @ ( F2 @ P6 ) )
          @ R2
          @ M3 ) ) ) ).

% mmap_fold_def
thf(fact_510_mmap__fold__def,axiom,
    ( mmap_f6516719976883110489_a_b_a
    = ( ^ [M3: list_P2992919015626629540_a_b_a,F2: produc7822699449245016212_a_b_a > produc7822699449245016212_a_b_a,C3: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,R2: list_P2992919015626629540_a_b_a] :
          ( foldl_1816153077079769819_a_b_a
          @ ^ [S2: list_P2992919015626629540_a_b_a,P6: produc7822699449245016212_a_b_a] :
              ( produc4579919614062918239_a_b_a
              @ ^ [K: a,V: mappin2365780136655028071_a_b_a] : ( mmap_c402980803363502225_a_b_a @ K @ V @ C3 @ S2 )
              @ ( F2 @ P6 ) )
          @ R2
          @ M3 ) ) ) ).

% mmap_fold_def
thf(fact_511_mmap__fold__def,axiom,
    ( mmap_fold_a_b
    = ( ^ [M3: list_P1467974523212632366od_a_b,F2: product_prod_a_b > product_prod_a_b,C3: b > b > b,R2: list_P1467974523212632366od_a_b] :
          ( foldl_4837193457258574491od_a_b
          @ ^ [S2: list_P1467974523212632366od_a_b,P6: product_prod_a_b] :
              ( produc611504906096598721od_a_b
              @ ^ [K: a,V: b] : ( mmap_combine_a_b @ K @ V @ C3 @ S2 )
              @ ( F2 @ P6 ) )
          @ R2
          @ M3 ) ) ) ).

% mmap_fold_def
thf(fact_512_mmap__fold__def,axiom,
    ( mmap_f8625263917883502961_d_nat
    = ( ^ [M3: list_P6684558232516339260_d_nat,F2: produc1990351791824763820_d_nat > produc1990351791824763820_d_nat,C3: produc5456268773499131775_d_nat > produc5456268773499131775_d_nat > produc5456268773499131775_d_nat,R2: list_P6684558232516339260_d_nat] :
          ( foldl_5754872400397474315_d_nat
          @ ^ [S2: list_P6684558232516339260_d_nat,P6: produc1990351791824763820_d_nat] :
              ( produc4938868912396122127_d_nat
              @ ^ [K: a,V: produc5456268773499131775_d_nat] : ( mmap_c2205071097579735977_d_nat @ K @ V @ C3 @ S2 )
              @ ( F2 @ P6 ) )
          @ R2
          @ M3 ) ) ) ).

% mmap_fold_def
thf(fact_513_the1__equality,axiom,
    ! [P: produc7822699449245016212_a_b_a > $o,A: produc7822699449245016212_a_b_a] :
      ( ? [X6: produc7822699449245016212_a_b_a] :
          ( ( P @ X6 )
          & ! [Y3: produc7822699449245016212_a_b_a] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ( P @ A )
       => ( ( the_Pr8288891123593341133_a_b_a @ P )
          = A ) ) ) ).

% the1_equality
thf(fact_514_the1__equality,axiom,
    ! [P: produc2062766750192416900_a_b_a > $o,A: produc2062766750192416900_a_b_a] :
      ( ? [X6: produc2062766750192416900_a_b_a] :
          ( ( P @ X6 )
          & ! [Y3: produc2062766750192416900_a_b_a] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ( P @ A )
       => ( ( the_Pr3099714871038386443_a_b_a @ P )
          = A ) ) ) ).

% the1_equality
thf(fact_515_the1__equality,axiom,
    ! [P: produc6343861673719254644_a_b_a > $o,A: produc6343861673719254644_a_b_a] :
      ( ? [X6: produc6343861673719254644_a_b_a] :
          ( ( P @ X6 )
          & ! [Y3: produc6343861673719254644_a_b_a] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ( P @ A )
       => ( ( the_Pr6227858779215488635_a_b_a @ P )
          = A ) ) ) ).

% the1_equality
thf(fact_516_the1__equality,axiom,
    ! [P: product_prod_a_b > $o,A: product_prod_a_b] :
      ( ? [X6: product_prod_a_b] :
          ( ( P @ X6 )
          & ! [Y3: product_prod_a_b] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ( P @ A )
       => ( ( the_Product_prod_a_b @ P )
          = A ) ) ) ).

% the1_equality
thf(fact_517_the1__equality,axiom,
    ! [P: produc1990351791824763820_d_nat > $o,A: produc1990351791824763820_d_nat] :
      ( ? [X6: produc1990351791824763820_d_nat] :
          ( ( P @ X6 )
          & ! [Y3: produc1990351791824763820_d_nat] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ( P @ A )
       => ( ( the_Pr6008708066028318565_d_nat @ P )
          = A ) ) ) ).

% the1_equality
thf(fact_518_the1I2,axiom,
    ! [P: produc7822699449245016212_a_b_a > $o,Q2: produc7822699449245016212_a_b_a > $o] :
      ( ? [X6: produc7822699449245016212_a_b_a] :
          ( ( P @ X6 )
          & ! [Y3: produc7822699449245016212_a_b_a] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ! [X3: produc7822699449245016212_a_b_a] :
            ( ( P @ X3 )
           => ( Q2 @ X3 ) )
       => ( Q2 @ ( the_Pr8288891123593341133_a_b_a @ P ) ) ) ) ).

% the1I2
thf(fact_519_the1I2,axiom,
    ! [P: produc2062766750192416900_a_b_a > $o,Q2: produc2062766750192416900_a_b_a > $o] :
      ( ? [X6: produc2062766750192416900_a_b_a] :
          ( ( P @ X6 )
          & ! [Y3: produc2062766750192416900_a_b_a] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ! [X3: produc2062766750192416900_a_b_a] :
            ( ( P @ X3 )
           => ( Q2 @ X3 ) )
       => ( Q2 @ ( the_Pr3099714871038386443_a_b_a @ P ) ) ) ) ).

% the1I2
thf(fact_520_the1I2,axiom,
    ! [P: produc6343861673719254644_a_b_a > $o,Q2: produc6343861673719254644_a_b_a > $o] :
      ( ? [X6: produc6343861673719254644_a_b_a] :
          ( ( P @ X6 )
          & ! [Y3: produc6343861673719254644_a_b_a] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ! [X3: produc6343861673719254644_a_b_a] :
            ( ( P @ X3 )
           => ( Q2 @ X3 ) )
       => ( Q2 @ ( the_Pr6227858779215488635_a_b_a @ P ) ) ) ) ).

% the1I2
thf(fact_521_the1I2,axiom,
    ! [P: product_prod_a_b > $o,Q2: product_prod_a_b > $o] :
      ( ? [X6: product_prod_a_b] :
          ( ( P @ X6 )
          & ! [Y3: product_prod_a_b] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ! [X3: product_prod_a_b] :
            ( ( P @ X3 )
           => ( Q2 @ X3 ) )
       => ( Q2 @ ( the_Product_prod_a_b @ P ) ) ) ) ).

% the1I2
thf(fact_522_the1I2,axiom,
    ! [P: produc1990351791824763820_d_nat > $o,Q2: produc1990351791824763820_d_nat > $o] :
      ( ? [X6: produc1990351791824763820_d_nat] :
          ( ( P @ X6 )
          & ! [Y3: produc1990351791824763820_d_nat] :
              ( ( P @ Y3 )
             => ( Y3 = X6 ) ) )
     => ( ! [X3: produc1990351791824763820_d_nat] :
            ( ( P @ X3 )
           => ( Q2 @ X3 ) )
       => ( Q2 @ ( the_Pr6008708066028318565_d_nat @ P ) ) ) ) ).

% the1I2
thf(fact_523_If__def,axiom,
    ( if_Pro7552986933024089678_a_b_a
    = ( ^ [P5: $o,X4: produc7822699449245016212_a_b_a,Y4: produc7822699449245016212_a_b_a] :
          ( the_Pr8288891123593341133_a_b_a
          @ ^ [Z3: produc7822699449245016212_a_b_a] :
              ( ( P5
               => ( Z3 = X4 ) )
              & ( ~ P5
               => ( Z3 = Y4 ) ) ) ) ) ) ).

% If_def
thf(fact_524_If__def,axiom,
    ( if_Pro5695710898613877322_a_b_a
    = ( ^ [P5: $o,X4: produc2062766750192416900_a_b_a,Y4: produc2062766750192416900_a_b_a] :
          ( the_Pr3099714871038386443_a_b_a
          @ ^ [Z3: produc2062766750192416900_a_b_a] :
              ( ( P5
               => ( Z3 = X4 ) )
              & ( ~ P5
               => ( Z3 = Y4 ) ) ) ) ) ) ).

% If_def
thf(fact_525_If__def,axiom,
    ( if_Pro8612401522923744058_a_b_a
    = ( ^ [P5: $o,X4: produc6343861673719254644_a_b_a,Y4: produc6343861673719254644_a_b_a] :
          ( the_Pr6227858779215488635_a_b_a
          @ ^ [Z3: produc6343861673719254644_a_b_a] :
              ( ( P5
               => ( Z3 = X4 ) )
              & ( ~ P5
               => ( Z3 = Y4 ) ) ) ) ) ) ).

% If_def
thf(fact_526_If__def,axiom,
    ( if_Product_prod_a_b
    = ( ^ [P5: $o,X4: product_prod_a_b,Y4: product_prod_a_b] :
          ( the_Product_prod_a_b
          @ ^ [Z3: product_prod_a_b] :
              ( ( P5
               => ( Z3 = X4 ) )
              & ( ~ P5
               => ( Z3 = Y4 ) ) ) ) ) ) ).

% If_def
thf(fact_527_If__def,axiom,
    ( if_Pro3152534550563304550_d_nat
    = ( ^ [P5: $o,X4: produc1990351791824763820_d_nat,Y4: produc1990351791824763820_d_nat] :
          ( the_Pr6008708066028318565_d_nat
          @ ^ [Z3: produc1990351791824763820_d_nat] :
              ( ( P5
               => ( Z3 = X4 ) )
              & ( ~ P5
               => ( Z3 = Y4 ) ) ) ) ) ) ).

% If_def
thf(fact_528_list__ex1__simps_I1_J,axiom,
    ! [P: product_prod_a_b > $o] :
      ~ ( list_e7533892414136369393od_a_b @ P @ nil_Product_prod_a_b ) ).

% list_ex1_simps(1)
thf(fact_529_list__ex1__simps_I1_J,axiom,
    ! [P: mappin2365780136655028071_a_b_a > $o] :
      ~ ( list_e8373548725095080158_a_b_a @ P @ nil_ma546712388551201761_a_b_a ) ).

% list_ex1_simps(1)
thf(fact_530_list__ex1__simps_I1_J,axiom,
    ! [P: list_P6684558232516339260_d_nat > $o] :
      ~ ( list_e6093811119172046259_d_nat @ P @ nil_li1008261075667615414_d_nat ) ).

% list_ex1_simps(1)
thf(fact_531_list__ex1__simps_I1_J,axiom,
    ! [P: b > $o] :
      ~ ( list_ex1_b @ P @ nil_b ) ).

% list_ex1_simps(1)
thf(fact_532_list__ex1__simps_I1_J,axiom,
    ! [P: a > $o] :
      ~ ( list_ex1_a @ P @ nil_a ) ).

% list_ex1_simps(1)
thf(fact_533_map__entry__code,axiom,
    ( map_en8183204059568693808_b_a_a
    = ( ^ [K: produc7822699449245016212_a_b_a,F2: a > a,M3: mappin4996136453398522101_b_a_a] :
          ( case_o5158257831078247380_a_a_a @ M3
          @ ^ [V: a] : ( update5685058447907601597_b_a_a @ K @ ( F2 @ V ) @ M3 )
          @ ( lookup1409243802988840364_b_a_a @ M3 @ K ) ) ) ) ).

% map_entry_code
thf(fact_534_map__entry__code,axiom,
    ( map_en7372027836849589853_a_b_a
    = ( ^ [K: list_P6684558232516339260_d_nat,F2: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,M3: mappin187059948859696622_a_b_a] :
          ( case_o4952957746345663458_a_b_a @ M3
          @ ^ [V: mappin2365780136655028071_a_b_a] : ( update5108779570945132176_a_b_a @ K @ ( F2 @ V ) @ M3 )
          @ ( lookup4661578797684577249_a_b_a @ M3 @ K ) ) ) ) ).

% map_entry_code
thf(fact_535_map__entry__code,axiom,
    ( map_en1595895783147578181_d_nat
    = ( ^ [K: a,F2: produc5456268773499131775_d_nat > produc5456268773499131775_d_nat,M3: mappin6535725686296717250_d_nat] :
          ( case_o6613947317223984502_d_nat @ M3
          @ ^ [V: produc5456268773499131775_d_nat] : ( update6116250764560905298_d_nat @ K @ ( F2 @ V ) @ M3 )
          @ ( lookup8480750236328838337_d_nat @ M3 @ K ) ) ) ) ).

% map_entry_code
thf(fact_536_map__entry__code,axiom,
    ( map_en8511718966582312493_a_b_a
    = ( ^ [K: a,F2: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,M3: mappin4519906018992269226_a_b_a] :
          ( case_o3188617522448952006_a_b_a @ M3
          @ ^ [V: mappin2365780136655028071_a_b_a] : ( update5632971445410033082_a_b_a @ K @ ( F2 @ V ) @ M3 )
          @ ( lookup1261341372242361769_a_b_a @ M3 @ K ) ) ) ) ).

% map_entry_code
thf(fact_537_map__entry__code,axiom,
    ( map_entry_a_b
    = ( ^ [K: a,F2: b > b,M3: mapping_a_b] :
          ( case_o5102372520203395616_a_b_b @ M3
          @ ^ [V: b] : ( update_a_b @ K @ ( F2 @ V ) @ M3 )
          @ ( lookup_a_b @ M3 @ K ) ) ) ) ).

% map_entry_code
thf(fact_538_map__entry__code,axiom,
    ( map_en8135975850489418638_a_b_a
    = ( ^ [K: product_prod_a_b,F2: a > a,M3: mappin2365780136655028071_a_b_a] :
          ( case_o2154183385986904842_b_a_a @ M3
          @ ^ [V: a] : ( update751575294834384705_a_b_a @ K @ ( F2 @ V ) @ M3 )
          @ ( lookup863726980359885586_a_b_a @ M3 @ K ) ) ) ) ).

% map_entry_code
thf(fact_539_old_Orec__unit__def,axiom,
    ( produc3610065700348839579_a_b_a
    = ( ^ [F12: produc7822699449245016212_a_b_a,X4: product_unit] : ( the_Pr8288891123593341133_a_b_a @ ( produc204438028682576217_a_b_a @ F12 @ X4 ) ) ) ) ).

% old.rec_unit_def
thf(fact_540_old_Orec__unit__def,axiom,
    ( produc1532443900092944829_a_b_a
    = ( ^ [F12: produc2062766750192416900_a_b_a,X4: product_unit] : ( the_Pr3099714871038386443_a_b_a @ ( produc2542501932168100223_a_b_a @ F12 @ X4 ) ) ) ) ).

% old.rec_unit_def
thf(fact_541_old_Orec__unit__def,axiom,
    ( produc5974643501638441005_a_b_a
    = ( ^ [F12: produc6343861673719254644_a_b_a,X4: product_unit] : ( the_Pr6227858779215488635_a_b_a @ ( produc5634096211792251119_a_b_a @ F12 @ X4 ) ) ) ) ).

% old.rec_unit_def
thf(fact_542_old_Orec__unit__def,axiom,
    ( produc7411352351530942817od_a_b
    = ( ^ [F12: product_prod_a_b,X4: product_unit] : ( the_Product_prod_a_b @ ( produc697149875024648227od_a_b @ F12 @ X4 ) ) ) ) ).

% old.rec_unit_def
thf(fact_543_old_Orec__unit__def,axiom,
    ( produc6778245085367883315_d_nat
    = ( ^ [F12: produc1990351791824763820_d_nat,X4: product_unit] : ( the_Pr6008708066028318565_d_nat @ ( produc3644926939410588145_d_nat @ F12 @ X4 ) ) ) ) ).

% old.rec_unit_def
thf(fact_544_product_Osimps_I1_J,axiom,
    ! [Uu2: list_b] :
      ( ( product_a_b @ nil_a @ Uu2 )
      = nil_Product_prod_a_b ) ).

% product.simps(1)
thf(fact_545_Nitpick_OThe__psimp,axiom,
    ! [P: produc7822699449245016212_a_b_a > $o,X: produc7822699449245016212_a_b_a] :
      ( ( P
        = ( ^ [Y: produc7822699449245016212_a_b_a,Z: produc7822699449245016212_a_b_a] : ( Y = Z )
          @ X ) )
     => ( ( the_Pr8288891123593341133_a_b_a @ P )
        = X ) ) ).

% Nitpick.The_psimp
thf(fact_546_Nitpick_OThe__psimp,axiom,
    ! [P: produc2062766750192416900_a_b_a > $o,X: produc2062766750192416900_a_b_a] :
      ( ( P
        = ( ^ [Y: produc2062766750192416900_a_b_a,Z: produc2062766750192416900_a_b_a] : ( Y = Z )
          @ X ) )
     => ( ( the_Pr3099714871038386443_a_b_a @ P )
        = X ) ) ).

% Nitpick.The_psimp
thf(fact_547_Nitpick_OThe__psimp,axiom,
    ! [P: produc6343861673719254644_a_b_a > $o,X: produc6343861673719254644_a_b_a] :
      ( ( P
        = ( ^ [Y: produc6343861673719254644_a_b_a,Z: produc6343861673719254644_a_b_a] : ( Y = Z )
          @ X ) )
     => ( ( the_Pr6227858779215488635_a_b_a @ P )
        = X ) ) ).

% Nitpick.The_psimp
thf(fact_548_Nitpick_OThe__psimp,axiom,
    ! [P: product_prod_a_b > $o,X: product_prod_a_b] :
      ( ( P
        = ( ^ [Y: product_prod_a_b,Z: product_prod_a_b] : ( Y = Z )
          @ X ) )
     => ( ( the_Product_prod_a_b @ P )
        = X ) ) ).

% Nitpick.The_psimp
thf(fact_549_Nitpick_OThe__psimp,axiom,
    ! [P: produc1990351791824763820_d_nat > $o,X: produc1990351791824763820_d_nat] :
      ( ( P
        = ( ^ [Y: produc1990351791824763820_d_nat,Z: produc1990351791824763820_d_nat] : ( Y = Z )
          @ X ) )
     => ( ( the_Pr6008708066028318565_d_nat @ P )
        = X ) ) ).

% Nitpick.The_psimp
thf(fact_550_ord_OLeast__def,axiom,
    ( least_8324104024763129024_a_b_a
    = ( ^ [Less_eq: produc7822699449245016212_a_b_a > produc7822699449245016212_a_b_a > $o,P5: produc7822699449245016212_a_b_a > $o] :
          ( the_Pr8288891123593341133_a_b_a
          @ ^ [X4: produc7822699449245016212_a_b_a] :
              ( ( P5 @ X4 )
              & ! [Y4: produc7822699449245016212_a_b_a] :
                  ( ( P5 @ Y4 )
                 => ( Less_eq @ X4 @ Y4 ) ) ) ) ) ) ).

% ord.Least_def
thf(fact_551_ord_OLeast__def,axiom,
    ( least_948633368502461784_a_b_a
    = ( ^ [Less_eq: produc2062766750192416900_a_b_a > produc2062766750192416900_a_b_a > $o,P5: produc2062766750192416900_a_b_a > $o] :
          ( the_Pr3099714871038386443_a_b_a
          @ ^ [X4: produc2062766750192416900_a_b_a] :
              ( ( P5 @ X4 )
              & ! [Y4: produc2062766750192416900_a_b_a] :
                  ( ( P5 @ Y4 )
                 => ( Less_eq @ X4 @ Y4 ) ) ) ) ) ) ).

% ord.Least_def
thf(fact_552_ord_OLeast__def,axiom,
    ( least_1923190301137237320_a_b_a
    = ( ^ [Less_eq: produc6343861673719254644_a_b_a > produc6343861673719254644_a_b_a > $o,P5: produc6343861673719254644_a_b_a > $o] :
          ( the_Pr6227858779215488635_a_b_a
          @ ^ [X4: produc6343861673719254644_a_b_a] :
              ( ( P5 @ X4 )
              & ! [Y4: produc6343861673719254644_a_b_a] :
                  ( ( P5 @ Y4 )
                 => ( Less_eq @ X4 @ Y4 ) ) ) ) ) ) ).

% ord.Least_def
thf(fact_553_ord_OLeast__def,axiom,
    ( least_4859807381444992892od_a_b
    = ( ^ [Less_eq: product_prod_a_b > product_prod_a_b > $o,P5: product_prod_a_b > $o] :
          ( the_Product_prod_a_b
          @ ^ [X4: product_prod_a_b] :
              ( ( P5 @ X4 )
              & ! [Y4: product_prod_a_b] :
                  ( ( P5 @ Y4 )
                 => ( Less_eq @ X4 @ Y4 ) ) ) ) ) ) ).

% ord.Least_def
thf(fact_554_ord_OLeast__def,axiom,
    ( least_4174451831939367384_d_nat
    = ( ^ [Less_eq: produc1990351791824763820_d_nat > produc1990351791824763820_d_nat > $o,P5: produc1990351791824763820_d_nat > $o] :
          ( the_Pr6008708066028318565_d_nat
          @ ^ [X4: produc1990351791824763820_d_nat] :
              ( ( P5 @ X4 )
              & ! [Y4: produc1990351791824763820_d_nat] :
                  ( ( P5 @ Y4 )
                 => ( Less_eq @ X4 @ Y4 ) ) ) ) ) ) ).

% ord.Least_def
thf(fact_555_mmap__combine_Osimps_I1_J,axiom,
    ! [K2: a,V2: mappin4996136453398522101_b_a_a,C2: mappin4996136453398522101_b_a_a > mappin4996136453398522101_b_a_a > mappin4996136453398522101_b_a_a] :
      ( ( mmap_c6646943743142367051_b_a_a @ K2 @ V2 @ C2 @ nil_Pr5740214961507490724_b_a_a )
      = ( cons_P220927179099053908_b_a_a @ ( produc4259011374121840526_b_a_a @ K2 @ V2 ) @ nil_Pr5740214961507490724_b_a_a ) ) ).

% mmap_combine.simps(1)
thf(fact_556_mmap__combine_Osimps_I1_J,axiom,
    ! [K2: a,V2: list_P6684558232516339260_d_nat,C2: list_P6684558232516339260_d_nat > list_P6684558232516339260_d_nat > list_P6684558232516339260_d_nat] :
      ( ( mmap_c2696983453546349414_d_nat @ K2 @ V2 @ C2 @ nil_Pr1577584438861778787_d_nat )
      = ( cons_P260773292763473843_d_nat @ ( produc919746265332163299_d_nat @ K2 @ V2 ) @ nil_Pr1577584438861778787_d_nat ) ) ).

% mmap_combine.simps(1)
thf(fact_557_mmap__combine_Osimps_I1_J,axiom,
    ! [K2: a,V2: list_P2992919015626629540_a_b_a,C2: list_P2992919015626629540_a_b_a > list_P2992919015626629540_a_b_a > list_P2992919015626629540_a_b_a] :
      ( ( mmap_c13429169768989646_a_b_a @ K2 @ V2 @ C2 @ nil_Pr7283556224815412427_a_b_a )
      = ( cons_P2675685553303450907_a_b_a @ ( produc6994869176599099851_a_b_a @ K2 @ V2 ) @ nil_Pr7283556224815412427_a_b_a ) ) ).

% mmap_combine.simps(1)
thf(fact_558_mmap__combine_Osimps_I1_J,axiom,
    ! [K2: a,V2: list_P1467974523212632366od_a_b,C2: list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b] :
      ( ( mmap_c5556936107559350148od_a_b @ K2 @ V2 @ C2 @ nil_Pr7243433834293086045od_a_b )
      = ( cons_P8483812562729421581od_a_b @ ( produc6643264353547226055od_a_b @ K2 @ V2 ) @ nil_Pr7243433834293086045od_a_b ) ) ).

% mmap_combine.simps(1)
thf(fact_559_mmap__combine_Osimps_I1_J,axiom,
    ! [K2: a,V2: a,C2: a > a > a] :
      ( ( mmap_combine_a_a @ K2 @ V2 @ C2 @ nil_Product_prod_a_a )
      = ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ K2 @ V2 ) @ nil_Product_prod_a_a ) ) ).

% mmap_combine.simps(1)
thf(fact_560_mmap__combine_Osimps_I1_J,axiom,
    ! [K2: a,V2: produc5456268773499131775_d_nat,C2: produc5456268773499131775_d_nat > produc5456268773499131775_d_nat > produc5456268773499131775_d_nat] :
      ( ( mmap_c2205071097579735977_d_nat @ K2 @ V2 @ C2 @ nil_Pr2300798825815202086_d_nat )
      = ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K2 @ V2 ) @ nil_Pr2300798825815202086_d_nat ) ) ).

% mmap_combine.simps(1)
thf(fact_561_mmap__combine_Osimps_I1_J,axiom,
    ! [K2: a,V2: b,C2: b > b > b] :
      ( ( mmap_combine_a_b @ K2 @ V2 @ C2 @ nil_Product_prod_a_b )
      = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ nil_Product_prod_a_b ) ) ).

% mmap_combine.simps(1)
thf(fact_562_mmap__combine_Osimps_I1_J,axiom,
    ! [K2: produc1990351791824763820_d_nat,V2: mappin2365780136655028071_a_b_a,C2: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a] :
      ( ( mmap_c5580072897124408041_a_b_a @ K2 @ V2 @ C2 @ nil_Pr3763312433997567610_a_b_a )
      = ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K2 @ V2 ) @ nil_Pr3763312433997567610_a_b_a ) ) ).

% mmap_combine.simps(1)
thf(fact_563_mmap__combine_Osimps_I1_J,axiom,
    ! [K2: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a,C2: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a] :
      ( ( mmap_c8213366964327184121_a_b_a @ K2 @ V2 @ C2 @ nil_Pr1908111780246848906_a_b_a )
      = ( cons_P2804281896780507450_a_b_a @ ( produc2616710729536263484_a_b_a @ K2 @ V2 ) @ nil_Pr1908111780246848906_a_b_a ) ) ).

% mmap_combine.simps(1)
thf(fact_564_mmap__combine_Osimps_I1_J,axiom,
    ! [K2: a,V2: mappin2365780136655028071_a_b_a,C2: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a] :
      ( ( mmap_c402980803363502225_a_b_a @ K2 @ V2 @ C2 @ nil_Pr7632893056509733646_a_b_a )
      = ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K2 @ V2 ) @ nil_Pr7632893056509733646_a_b_a ) ) ).

% mmap_combine.simps(1)
thf(fact_565_old_Orec__bool__def,axiom,
    ( produc68458625753171739_a_b_a
    = ( ^ [F12: produc7822699449245016212_a_b_a,F23: produc7822699449245016212_a_b_a,X4: $o] : ( the_Pr8288891123593341133_a_b_a @ ( produc6310648794561543909_a_b_a @ F12 @ F23 @ X4 ) ) ) ) ).

% old.rec_bool_def
thf(fact_566_old_Orec__bool__def,axiom,
    ( produc9154551198695911741_a_b_a
    = ( ^ [F12: produc2062766750192416900_a_b_a,F23: produc2062766750192416900_a_b_a,X4: $o] : ( the_Pr3099714871038386443_a_b_a @ ( produc6585200446582009075_a_b_a @ F12 @ F23 @ X4 ) ) ) ) ).

% old.rec_bool_def
thf(fact_567_old_Orec__bool__def,axiom,
    ( produc176917111123965869_a_b_a
    = ( ^ [F12: produc6343861673719254644_a_b_a,F23: produc6343861673719254644_a_b_a,X4: $o] : ( the_Pr6227858779215488635_a_b_a @ ( produc6018843065701860963_a_b_a @ F12 @ F23 @ X4 ) ) ) ) ).

% old.rec_bool_def
thf(fact_568_old_Orec__bool__def,axiom,
    ( produc8670534973760119009od_a_b
    = ( ^ [F12: product_prod_a_b,F23: product_prod_a_b,X4: $o] : ( the_Product_prod_a_b @ ( produc5869776753928712599od_a_b @ F12 @ F23 @ X4 ) ) ) ) ).

% old.rec_bool_def
thf(fact_569_old_Orec__bool__def,axiom,
    ( produc8752953327808845491_d_nat
    = ( ^ [F12: produc1990351791824763820_d_nat,F23: produc1990351791824763820_d_nat,X4: $o] : ( the_Pr6008708066028318565_d_nat @ ( produc7336012173783415165_d_nat @ F12 @ F23 @ X4 ) ) ) ) ).

% old.rec_bool_def
thf(fact_570_list_Oinject,axiom,
    ! [X21: product_prod_a_b,X222: list_P1467974523212632366od_a_b,Y21: product_prod_a_b,Y22: list_P1467974523212632366od_a_b] :
      ( ( ( cons_P7316939131009794654od_a_b @ X21 @ X222 )
        = ( cons_P7316939131009794654od_a_b @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_571_list_Oinject,axiom,
    ! [X21: mappin2365780136655028071_a_b_a,X222: list_m9210641325446214647_a_b_a,Y21: mappin2365780136655028071_a_b_a,Y22: list_m9210641325446214647_a_b_a] :
      ( ( ( cons_m8574943908289518641_a_b_a @ X21 @ X222 )
        = ( cons_m8574943908289518641_a_b_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_572_list_Oinject,axiom,
    ! [X21: list_P6684558232516339260_d_nat,X222: list_l337834041258660428_d_nat,Y21: list_P6684558232516339260_d_nat,Y22: list_l337834041258660428_d_nat] :
      ( ( ( cons_l4410424432949683974_d_nat @ X21 @ X222 )
        = ( cons_l4410424432949683974_d_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_573_list_Oinject,axiom,
    ! [X21: b,X222: list_b,Y21: b,Y22: list_b] :
      ( ( ( cons_b @ X21 @ X222 )
        = ( cons_b @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_574_list_Oinject,axiom,
    ! [X21: a,X222: list_a,Y21: a,Y22: list_a] :
      ( ( ( cons_a @ X21 @ X222 )
        = ( cons_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_575_transpose_Ocases,axiom,
    ! [X: list_l8778361445553822004od_a_b] :
      ( ( X != nil_li8635777946366267444od_a_b )
     => ( ! [Xss: list_l8778361445553822004od_a_b] :
            ( X
           != ( cons_l907386227682280420od_a_b @ nil_Product_prod_a_b @ Xss ) )
       => ~ ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Xss: list_l8778361445553822004od_a_b] :
              ( X
             != ( cons_l907386227682280420od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_576_transpose_Ocases,axiom,
    ! [X: list_l3040532949214407175_a_b_a] :
      ( ( X != nil_li7899906211823649649_a_b_a )
     => ( ! [Xss: list_l3040532949214407175_a_b_a] :
            ( X
           != ( cons_l8438923613303137217_a_b_a @ nil_ma546712388551201761_a_b_a @ Xss ) )
       => ~ ! [X3: mappin2365780136655028071_a_b_a,Xs: list_m9210641325446214647_a_b_a,Xss: list_l3040532949214407175_a_b_a] :
              ( X
             != ( cons_l8438923613303137217_a_b_a @ ( cons_m8574943908289518641_a_b_a @ X3 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_577_transpose_Ocases,axiom,
    ! [X: list_l5666407289914589148_d_nat] :
      ( ( X != nil_li4515601735571940806_d_nat )
     => ( ! [Xss: list_l5666407289914589148_d_nat] :
            ( X
           != ( cons_l8566576933256865302_d_nat @ nil_li1008261075667615414_d_nat @ Xss ) )
       => ~ ! [X3: list_P6684558232516339260_d_nat,Xs: list_l337834041258660428_d_nat,Xss: list_l5666407289914589148_d_nat] :
              ( X
             != ( cons_l8566576933256865302_d_nat @ ( cons_l4410424432949683974_d_nat @ X3 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_578_transpose_Ocases,axiom,
    ! [X: list_list_b] :
      ( ( X != nil_list_b )
     => ( ! [Xss: list_list_b] :
            ( X
           != ( cons_list_b @ nil_b @ Xss ) )
       => ~ ! [X3: b,Xs: list_b,Xss: list_list_b] :
              ( X
             != ( cons_list_b @ ( cons_b @ X3 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_579_transpose_Ocases,axiom,
    ! [X: list_list_a] :
      ( ( X != nil_list_a )
     => ( ! [Xss: list_list_a] :
            ( X
           != ( cons_list_a @ nil_a @ Xss ) )
       => ~ ! [X3: a,Xs: list_a,Xss: list_list_a] :
              ( X
             != ( cons_list_a @ ( cons_a @ X3 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_580_transpose_Ocases,axiom,
    ! [X: list_l337834041258660428_d_nat] :
      ( ( X != nil_li1008261075667615414_d_nat )
     => ( ! [Xss: list_l337834041258660428_d_nat] :
            ( X
           != ( cons_l4410424432949683974_d_nat @ nil_Pr2300798825815202086_d_nat @ Xss ) )
       => ~ ! [X3: produc1990351791824763820_d_nat,Xs: list_P6684558232516339260_d_nat,Xss: list_l337834041258660428_d_nat] :
              ( X
             != ( cons_l4410424432949683974_d_nat @ ( cons_P2594832900118292854_d_nat @ X3 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_581_not__Cons__self2,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( cons_P7316939131009794654od_a_b @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_582_not__Cons__self2,axiom,
    ! [X: mappin2365780136655028071_a_b_a,Xs2: list_m9210641325446214647_a_b_a] :
      ( ( cons_m8574943908289518641_a_b_a @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_583_not__Cons__self2,axiom,
    ! [X: list_P6684558232516339260_d_nat,Xs2: list_l337834041258660428_d_nat] :
      ( ( cons_l4410424432949683974_d_nat @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_584_not__Cons__self2,axiom,
    ! [X: b,Xs2: list_b] :
      ( ( cons_b @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_585_not__Cons__self2,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( cons_a @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_586_splice_Ocases,axiom,
    ! [X: produc7397007428618769063od_a_b] :
      ( ! [Ys: list_P1467974523212632366od_a_b] :
          ( X
         != ( produc5288560138351607191od_a_b @ nil_Product_prod_a_b @ Ys ) )
     => ~ ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
            ( X
           != ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ Ys ) ) ) ).

% splice.cases
thf(fact_587_splice_Ocases,axiom,
    ! [X: produc3116146679011544271_a_b_a] :
      ( ! [Ys: list_m9210641325446214647_a_b_a] :
          ( X
         != ( produc6824944014479491719_a_b_a @ nil_ma546712388551201761_a_b_a @ Ys ) )
     => ~ ! [X3: mappin2365780136655028071_a_b_a,Xs: list_m9210641325446214647_a_b_a,Ys: list_m9210641325446214647_a_b_a] :
            ( X
           != ( produc6824944014479491719_a_b_a @ ( cons_m8574943908289518641_a_b_a @ X3 @ Xs ) @ Ys ) ) ) ).

% splice.cases
thf(fact_588_splice_Ocases,axiom,
    ! [X: produc7045765337849118585_d_nat] :
      ( ! [Ys: list_l337834041258660428_d_nat] :
          ( X
         != ( produc4511192556619310001_d_nat @ nil_li1008261075667615414_d_nat @ Ys ) )
     => ~ ! [X3: list_P6684558232516339260_d_nat,Xs: list_l337834041258660428_d_nat,Ys: list_l337834041258660428_d_nat] :
            ( X
           != ( produc4511192556619310001_d_nat @ ( cons_l4410424432949683974_d_nat @ X3 @ Xs ) @ Ys ) ) ) ).

% splice.cases
thf(fact_589_splice_Ocases,axiom,
    ! [X: produc3963297410138542439list_b] :
      ( ! [Ys: list_b] :
          ( X
         != ( produc1564554178308465111list_b @ nil_b @ Ys ) )
     => ~ ! [X3: b,Xs: list_b,Ys: list_b] :
            ( X
           != ( produc1564554178308465111list_b @ ( cons_b @ X3 @ Xs ) @ Ys ) ) ) ).

% splice.cases
thf(fact_590_splice_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [Ys: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ nil_a @ Ys ) )
     => ~ ! [X3: a,Xs: list_a,Ys: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs ) @ Ys ) ) ) ).

% splice.cases
thf(fact_591_shuffles_Ocases,axiom,
    ! [X: produc7397007428618769063od_a_b] :
      ( ! [Ys: list_P1467974523212632366od_a_b] :
          ( X
         != ( produc5288560138351607191od_a_b @ nil_Product_prod_a_b @ Ys ) )
     => ( ! [Xs: list_P1467974523212632366od_a_b] :
            ( X
           != ( produc5288560138351607191od_a_b @ Xs @ nil_Product_prod_a_b ) )
       => ~ ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Y3: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
              ( X
             != ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ ( cons_P7316939131009794654od_a_b @ Y3 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_592_shuffles_Ocases,axiom,
    ! [X: produc3116146679011544271_a_b_a] :
      ( ! [Ys: list_m9210641325446214647_a_b_a] :
          ( X
         != ( produc6824944014479491719_a_b_a @ nil_ma546712388551201761_a_b_a @ Ys ) )
     => ( ! [Xs: list_m9210641325446214647_a_b_a] :
            ( X
           != ( produc6824944014479491719_a_b_a @ Xs @ nil_ma546712388551201761_a_b_a ) )
       => ~ ! [X3: mappin2365780136655028071_a_b_a,Xs: list_m9210641325446214647_a_b_a,Y3: mappin2365780136655028071_a_b_a,Ys: list_m9210641325446214647_a_b_a] :
              ( X
             != ( produc6824944014479491719_a_b_a @ ( cons_m8574943908289518641_a_b_a @ X3 @ Xs ) @ ( cons_m8574943908289518641_a_b_a @ Y3 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_593_shuffles_Ocases,axiom,
    ! [X: produc7045765337849118585_d_nat] :
      ( ! [Ys: list_l337834041258660428_d_nat] :
          ( X
         != ( produc4511192556619310001_d_nat @ nil_li1008261075667615414_d_nat @ Ys ) )
     => ( ! [Xs: list_l337834041258660428_d_nat] :
            ( X
           != ( produc4511192556619310001_d_nat @ Xs @ nil_li1008261075667615414_d_nat ) )
       => ~ ! [X3: list_P6684558232516339260_d_nat,Xs: list_l337834041258660428_d_nat,Y3: list_P6684558232516339260_d_nat,Ys: list_l337834041258660428_d_nat] :
              ( X
             != ( produc4511192556619310001_d_nat @ ( cons_l4410424432949683974_d_nat @ X3 @ Xs ) @ ( cons_l4410424432949683974_d_nat @ Y3 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_594_shuffles_Ocases,axiom,
    ! [X: produc3963297410138542439list_b] :
      ( ! [Ys: list_b] :
          ( X
         != ( produc1564554178308465111list_b @ nil_b @ Ys ) )
     => ( ! [Xs: list_b] :
            ( X
           != ( produc1564554178308465111list_b @ Xs @ nil_b ) )
       => ~ ! [X3: b,Xs: list_b,Y3: b,Ys: list_b] :
              ( X
             != ( produc1564554178308465111list_b @ ( cons_b @ X3 @ Xs ) @ ( cons_b @ Y3 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_595_shuffles_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [Ys: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ nil_a @ Ys ) )
     => ( ! [Xs: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ Xs @ nil_a ) )
       => ~ ! [X3: a,Xs: list_a,Y3: a,Ys: list_a] :
              ( X
             != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs ) @ ( cons_a @ Y3 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_596_sorted__wrt_Ocases,axiom,
    ! [X: produc3334210663054469735od_a_b] :
      ( ! [P7: product_prod_a_b > product_prod_a_b > $o] :
          ( X
         != ( produc7704224619955794135od_a_b @ P7 @ nil_Product_prod_a_b ) )
     => ~ ! [P7: product_prod_a_b > product_prod_a_b > $o,X3: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
            ( X
           != ( produc7704224619955794135od_a_b @ P7 @ ( cons_P7316939131009794654od_a_b @ X3 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_597_sorted__wrt_Ocases,axiom,
    ! [X: produc982609575658243294_a_b_a] :
      ( ! [P7: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a > $o] :
          ( X
         != ( produc4941649227406914392_a_b_a @ P7 @ nil_ma546712388551201761_a_b_a ) )
     => ~ ! [P7: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a > $o,X3: mappin2365780136655028071_a_b_a,Ys: list_m9210641325446214647_a_b_a] :
            ( X
           != ( produc4941649227406914392_a_b_a @ P7 @ ( cons_m8574943908289518641_a_b_a @ X3 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_598_sorted__wrt_Ocases,axiom,
    ! [X: produc4000265523682700425_d_nat] :
      ( ! [P7: list_P6684558232516339260_d_nat > list_P6684558232516339260_d_nat > $o] :
          ( X
         != ( produc2760578415356222339_d_nat @ P7 @ nil_li1008261075667615414_d_nat ) )
     => ~ ! [P7: list_P6684558232516339260_d_nat > list_P6684558232516339260_d_nat > $o,X3: list_P6684558232516339260_d_nat,Ys: list_l337834041258660428_d_nat] :
            ( X
           != ( produc2760578415356222339_d_nat @ P7 @ ( cons_l4410424432949683974_d_nat @ X3 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_599_sorted__wrt_Ocases,axiom,
    ! [X: produc5185152304234826110list_b] :
      ( ! [P7: b > b > $o] :
          ( X
         != ( produc8193136575784045678list_b @ P7 @ nil_b ) )
     => ~ ! [P7: b > b > $o,X3: b,Ys: list_b] :
            ( X
           != ( produc8193136575784045678list_b @ P7 @ ( cons_b @ X3 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_600_sorted__wrt_Ocases,axiom,
    ! [X: produc5032551385658279741list_a] :
      ( ! [P7: a > a > $o] :
          ( X
         != ( produc8111569692950616493list_a @ P7 @ nil_a ) )
     => ~ ! [P7: a > a > $o,X3: a,Ys: list_a] :
            ( X
           != ( produc8111569692950616493list_a @ P7 @ ( cons_a @ X3 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_601_successively_Ocases,axiom,
    ! [X: produc3334210663054469735od_a_b] :
      ( ! [P7: product_prod_a_b > product_prod_a_b > $o] :
          ( X
         != ( produc7704224619955794135od_a_b @ P7 @ nil_Product_prod_a_b ) )
     => ( ! [P7: product_prod_a_b > product_prod_a_b > $o,X3: product_prod_a_b] :
            ( X
           != ( produc7704224619955794135od_a_b @ P7 @ ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) ) )
       => ~ ! [P7: product_prod_a_b > product_prod_a_b > $o,X3: product_prod_a_b,Y3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
              ( X
             != ( produc7704224619955794135od_a_b @ P7 @ ( cons_P7316939131009794654od_a_b @ X3 @ ( cons_P7316939131009794654od_a_b @ Y3 @ Xs ) ) ) ) ) ) ).

% successively.cases
thf(fact_602_successively_Ocases,axiom,
    ! [X: produc982609575658243294_a_b_a] :
      ( ! [P7: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a > $o] :
          ( X
         != ( produc4941649227406914392_a_b_a @ P7 @ nil_ma546712388551201761_a_b_a ) )
     => ( ! [P7: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a > $o,X3: mappin2365780136655028071_a_b_a] :
            ( X
           != ( produc4941649227406914392_a_b_a @ P7 @ ( cons_m8574943908289518641_a_b_a @ X3 @ nil_ma546712388551201761_a_b_a ) ) )
       => ~ ! [P7: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a > $o,X3: mappin2365780136655028071_a_b_a,Y3: mappin2365780136655028071_a_b_a,Xs: list_m9210641325446214647_a_b_a] :
              ( X
             != ( produc4941649227406914392_a_b_a @ P7 @ ( cons_m8574943908289518641_a_b_a @ X3 @ ( cons_m8574943908289518641_a_b_a @ Y3 @ Xs ) ) ) ) ) ) ).

% successively.cases
thf(fact_603_successively_Ocases,axiom,
    ! [X: produc4000265523682700425_d_nat] :
      ( ! [P7: list_P6684558232516339260_d_nat > list_P6684558232516339260_d_nat > $o] :
          ( X
         != ( produc2760578415356222339_d_nat @ P7 @ nil_li1008261075667615414_d_nat ) )
     => ( ! [P7: list_P6684558232516339260_d_nat > list_P6684558232516339260_d_nat > $o,X3: list_P6684558232516339260_d_nat] :
            ( X
           != ( produc2760578415356222339_d_nat @ P7 @ ( cons_l4410424432949683974_d_nat @ X3 @ nil_li1008261075667615414_d_nat ) ) )
       => ~ ! [P7: list_P6684558232516339260_d_nat > list_P6684558232516339260_d_nat > $o,X3: list_P6684558232516339260_d_nat,Y3: list_P6684558232516339260_d_nat,Xs: list_l337834041258660428_d_nat] :
              ( X
             != ( produc2760578415356222339_d_nat @ P7 @ ( cons_l4410424432949683974_d_nat @ X3 @ ( cons_l4410424432949683974_d_nat @ Y3 @ Xs ) ) ) ) ) ) ).

% successively.cases
thf(fact_604_successively_Ocases,axiom,
    ! [X: produc5185152304234826110list_b] :
      ( ! [P7: b > b > $o] :
          ( X
         != ( produc8193136575784045678list_b @ P7 @ nil_b ) )
     => ( ! [P7: b > b > $o,X3: b] :
            ( X
           != ( produc8193136575784045678list_b @ P7 @ ( cons_b @ X3 @ nil_b ) ) )
       => ~ ! [P7: b > b > $o,X3: b,Y3: b,Xs: list_b] :
              ( X
             != ( produc8193136575784045678list_b @ P7 @ ( cons_b @ X3 @ ( cons_b @ Y3 @ Xs ) ) ) ) ) ) ).

% successively.cases
thf(fact_605_successively_Ocases,axiom,
    ! [X: produc5032551385658279741list_a] :
      ( ! [P7: a > a > $o] :
          ( X
         != ( produc8111569692950616493list_a @ P7 @ nil_a ) )
     => ( ! [P7: a > a > $o,X3: a] :
            ( X
           != ( produc8111569692950616493list_a @ P7 @ ( cons_a @ X3 @ nil_a ) ) )
       => ~ ! [P7: a > a > $o,X3: a,Y3: a,Xs: list_a] :
              ( X
             != ( produc8111569692950616493list_a @ P7 @ ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs ) ) ) ) ) ) ).

% successively.cases
thf(fact_606_list_Odistinct_I1_J,axiom,
    ! [X21: product_prod_a_b,X222: list_P1467974523212632366od_a_b] :
      ( nil_Product_prod_a_b
     != ( cons_P7316939131009794654od_a_b @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_607_list_Odistinct_I1_J,axiom,
    ! [X21: mappin2365780136655028071_a_b_a,X222: list_m9210641325446214647_a_b_a] :
      ( nil_ma546712388551201761_a_b_a
     != ( cons_m8574943908289518641_a_b_a @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_608_list_Odistinct_I1_J,axiom,
    ! [X21: list_P6684558232516339260_d_nat,X222: list_l337834041258660428_d_nat] :
      ( nil_li1008261075667615414_d_nat
     != ( cons_l4410424432949683974_d_nat @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_609_list_Odistinct_I1_J,axiom,
    ! [X21: b,X222: list_b] :
      ( nil_b
     != ( cons_b @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_610_list_Odistinct_I1_J,axiom,
    ! [X21: a,X222: list_a] :
      ( nil_a
     != ( cons_a @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_611_list_OdiscI,axiom,
    ! [List: list_P1467974523212632366od_a_b,X21: product_prod_a_b,X222: list_P1467974523212632366od_a_b] :
      ( ( List
        = ( cons_P7316939131009794654od_a_b @ X21 @ X222 ) )
     => ( List != nil_Product_prod_a_b ) ) ).

% list.discI
thf(fact_612_list_OdiscI,axiom,
    ! [List: list_m9210641325446214647_a_b_a,X21: mappin2365780136655028071_a_b_a,X222: list_m9210641325446214647_a_b_a] :
      ( ( List
        = ( cons_m8574943908289518641_a_b_a @ X21 @ X222 ) )
     => ( List != nil_ma546712388551201761_a_b_a ) ) ).

% list.discI
thf(fact_613_list_OdiscI,axiom,
    ! [List: list_l337834041258660428_d_nat,X21: list_P6684558232516339260_d_nat,X222: list_l337834041258660428_d_nat] :
      ( ( List
        = ( cons_l4410424432949683974_d_nat @ X21 @ X222 ) )
     => ( List != nil_li1008261075667615414_d_nat ) ) ).

% list.discI
thf(fact_614_list_OdiscI,axiom,
    ! [List: list_b,X21: b,X222: list_b] :
      ( ( List
        = ( cons_b @ X21 @ X222 ) )
     => ( List != nil_b ) ) ).

% list.discI
thf(fact_615_list_OdiscI,axiom,
    ! [List: list_a,X21: a,X222: list_a] :
      ( ( List
        = ( cons_a @ X21 @ X222 ) )
     => ( List != nil_a ) ) ).

% list.discI
thf(fact_616_list_Oexhaust,axiom,
    ! [Y5: list_P1467974523212632366od_a_b] :
      ( ( Y5 != nil_Product_prod_a_b )
     => ~ ! [X212: product_prod_a_b,X223: list_P1467974523212632366od_a_b] :
            ( Y5
           != ( cons_P7316939131009794654od_a_b @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_617_list_Oexhaust,axiom,
    ! [Y5: list_m9210641325446214647_a_b_a] :
      ( ( Y5 != nil_ma546712388551201761_a_b_a )
     => ~ ! [X212: mappin2365780136655028071_a_b_a,X223: list_m9210641325446214647_a_b_a] :
            ( Y5
           != ( cons_m8574943908289518641_a_b_a @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_618_list_Oexhaust,axiom,
    ! [Y5: list_l337834041258660428_d_nat] :
      ( ( Y5 != nil_li1008261075667615414_d_nat )
     => ~ ! [X212: list_P6684558232516339260_d_nat,X223: list_l337834041258660428_d_nat] :
            ( Y5
           != ( cons_l4410424432949683974_d_nat @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_619_list_Oexhaust,axiom,
    ! [Y5: list_b] :
      ( ( Y5 != nil_b )
     => ~ ! [X212: b,X223: list_b] :
            ( Y5
           != ( cons_b @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_620_list_Oexhaust,axiom,
    ! [Y5: list_a] :
      ( ( Y5 != nil_a )
     => ~ ! [X212: a,X223: list_a] :
            ( Y5
           != ( cons_a @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_621_remdups__adj_Ocases,axiom,
    ! [X: list_P1467974523212632366od_a_b] :
      ( ( X != nil_Product_prod_a_b )
     => ( ! [X3: product_prod_a_b] :
            ( X
           != ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) )
       => ~ ! [X3: product_prod_a_b,Y3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
              ( X
             != ( cons_P7316939131009794654od_a_b @ X3 @ ( cons_P7316939131009794654od_a_b @ Y3 @ Xs ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_622_remdups__adj_Ocases,axiom,
    ! [X: list_m9210641325446214647_a_b_a] :
      ( ( X != nil_ma546712388551201761_a_b_a )
     => ( ! [X3: mappin2365780136655028071_a_b_a] :
            ( X
           != ( cons_m8574943908289518641_a_b_a @ X3 @ nil_ma546712388551201761_a_b_a ) )
       => ~ ! [X3: mappin2365780136655028071_a_b_a,Y3: mappin2365780136655028071_a_b_a,Xs: list_m9210641325446214647_a_b_a] :
              ( X
             != ( cons_m8574943908289518641_a_b_a @ X3 @ ( cons_m8574943908289518641_a_b_a @ Y3 @ Xs ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_623_remdups__adj_Ocases,axiom,
    ! [X: list_l337834041258660428_d_nat] :
      ( ( X != nil_li1008261075667615414_d_nat )
     => ( ! [X3: list_P6684558232516339260_d_nat] :
            ( X
           != ( cons_l4410424432949683974_d_nat @ X3 @ nil_li1008261075667615414_d_nat ) )
       => ~ ! [X3: list_P6684558232516339260_d_nat,Y3: list_P6684558232516339260_d_nat,Xs: list_l337834041258660428_d_nat] :
              ( X
             != ( cons_l4410424432949683974_d_nat @ X3 @ ( cons_l4410424432949683974_d_nat @ Y3 @ Xs ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_624_remdups__adj_Ocases,axiom,
    ! [X: list_b] :
      ( ( X != nil_b )
     => ( ! [X3: b] :
            ( X
           != ( cons_b @ X3 @ nil_b ) )
       => ~ ! [X3: b,Y3: b,Xs: list_b] :
              ( X
             != ( cons_b @ X3 @ ( cons_b @ Y3 @ Xs ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_625_remdups__adj_Ocases,axiom,
    ! [X: list_a] :
      ( ( X != nil_a )
     => ( ! [X3: a] :
            ( X
           != ( cons_a @ X3 @ nil_a ) )
       => ~ ! [X3: a,Y3: a,Xs: list_a] :
              ( X
             != ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_626_neq__Nil__conv,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b] :
      ( ( Xs2 != nil_Product_prod_a_b )
      = ( ? [Y4: product_prod_a_b,Ys2: list_P1467974523212632366od_a_b] :
            ( Xs2
            = ( cons_P7316939131009794654od_a_b @ Y4 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_627_neq__Nil__conv,axiom,
    ! [Xs2: list_m9210641325446214647_a_b_a] :
      ( ( Xs2 != nil_ma546712388551201761_a_b_a )
      = ( ? [Y4: mappin2365780136655028071_a_b_a,Ys2: list_m9210641325446214647_a_b_a] :
            ( Xs2
            = ( cons_m8574943908289518641_a_b_a @ Y4 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_628_neq__Nil__conv,axiom,
    ! [Xs2: list_l337834041258660428_d_nat] :
      ( ( Xs2 != nil_li1008261075667615414_d_nat )
      = ( ? [Y4: list_P6684558232516339260_d_nat,Ys2: list_l337834041258660428_d_nat] :
            ( Xs2
            = ( cons_l4410424432949683974_d_nat @ Y4 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_629_neq__Nil__conv,axiom,
    ! [Xs2: list_b] :
      ( ( Xs2 != nil_b )
      = ( ? [Y4: b,Ys2: list_b] :
            ( Xs2
            = ( cons_b @ Y4 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_630_neq__Nil__conv,axiom,
    ! [Xs2: list_a] :
      ( ( Xs2 != nil_a )
      = ( ? [Y4: a,Ys2: list_a] :
            ( Xs2
            = ( cons_a @ Y4 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_631_list__induct2_H,axiom,
    ! [P: list_b > list_b > $o,Xs2: list_b,Ys3: list_b] :
      ( ( P @ nil_b @ nil_b )
     => ( ! [X3: b,Xs: list_b] : ( P @ ( cons_b @ X3 @ Xs ) @ nil_b )
       => ( ! [Y3: b,Ys: list_b] : ( P @ nil_b @ ( cons_b @ Y3 @ Ys ) )
         => ( ! [X3: b,Xs: list_b,Y3: b,Ys: list_b] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_b @ X3 @ Xs ) @ ( cons_b @ Y3 @ Ys ) ) )
           => ( P @ Xs2 @ Ys3 ) ) ) ) ) ).

% list_induct2'
thf(fact_632_list__induct2_H,axiom,
    ! [P: list_b > list_a > $o,Xs2: list_b,Ys3: list_a] :
      ( ( P @ nil_b @ nil_a )
     => ( ! [X3: b,Xs: list_b] : ( P @ ( cons_b @ X3 @ Xs ) @ nil_a )
       => ( ! [Y3: a,Ys: list_a] : ( P @ nil_b @ ( cons_a @ Y3 @ Ys ) )
         => ( ! [X3: b,Xs: list_b,Y3: a,Ys: list_a] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_b @ X3 @ Xs ) @ ( cons_a @ Y3 @ Ys ) ) )
           => ( P @ Xs2 @ Ys3 ) ) ) ) ) ).

% list_induct2'
thf(fact_633_list__induct2_H,axiom,
    ! [P: list_a > list_b > $o,Xs2: list_a,Ys3: list_b] :
      ( ( P @ nil_a @ nil_b )
     => ( ! [X3: a,Xs: list_a] : ( P @ ( cons_a @ X3 @ Xs ) @ nil_b )
       => ( ! [Y3: b,Ys: list_b] : ( P @ nil_a @ ( cons_b @ Y3 @ Ys ) )
         => ( ! [X3: a,Xs: list_a,Y3: b,Ys: list_b] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_a @ X3 @ Xs ) @ ( cons_b @ Y3 @ Ys ) ) )
           => ( P @ Xs2 @ Ys3 ) ) ) ) ) ).

% list_induct2'
thf(fact_634_list__induct2_H,axiom,
    ! [P: list_a > list_a > $o,Xs2: list_a,Ys3: list_a] :
      ( ( P @ nil_a @ nil_a )
     => ( ! [X3: a,Xs: list_a] : ( P @ ( cons_a @ X3 @ Xs ) @ nil_a )
       => ( ! [Y3: a,Ys: list_a] : ( P @ nil_a @ ( cons_a @ Y3 @ Ys ) )
         => ( ! [X3: a,Xs: list_a,Y3: a,Ys: list_a] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_a @ X3 @ Xs ) @ ( cons_a @ Y3 @ Ys ) ) )
           => ( P @ Xs2 @ Ys3 ) ) ) ) ) ).

% list_induct2'
thf(fact_635_list__induct2_H,axiom,
    ! [P: list_P1467974523212632366od_a_b > list_b > $o,Xs2: list_P1467974523212632366od_a_b,Ys3: list_b] :
      ( ( P @ nil_Product_prod_a_b @ nil_b )
     => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] : ( P @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ nil_b )
       => ( ! [Y3: b,Ys: list_b] : ( P @ nil_Product_prod_a_b @ ( cons_b @ Y3 @ Ys ) )
         => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Y3: b,Ys: list_b] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ ( cons_b @ Y3 @ Ys ) ) )
           => ( P @ Xs2 @ Ys3 ) ) ) ) ) ).

% list_induct2'
thf(fact_636_list__induct2_H,axiom,
    ! [P: list_P1467974523212632366od_a_b > list_a > $o,Xs2: list_P1467974523212632366od_a_b,Ys3: list_a] :
      ( ( P @ nil_Product_prod_a_b @ nil_a )
     => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] : ( P @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ nil_a )
       => ( ! [Y3: a,Ys: list_a] : ( P @ nil_Product_prod_a_b @ ( cons_a @ Y3 @ Ys ) )
         => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Y3: a,Ys: list_a] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ ( cons_a @ Y3 @ Ys ) ) )
           => ( P @ Xs2 @ Ys3 ) ) ) ) ) ).

% list_induct2'
thf(fact_637_list__induct2_H,axiom,
    ! [P: list_b > list_P1467974523212632366od_a_b > $o,Xs2: list_b,Ys3: list_P1467974523212632366od_a_b] :
      ( ( P @ nil_b @ nil_Product_prod_a_b )
     => ( ! [X3: b,Xs: list_b] : ( P @ ( cons_b @ X3 @ Xs ) @ nil_Product_prod_a_b )
       => ( ! [Y3: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] : ( P @ nil_b @ ( cons_P7316939131009794654od_a_b @ Y3 @ Ys ) )
         => ( ! [X3: b,Xs: list_b,Y3: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_b @ X3 @ Xs ) @ ( cons_P7316939131009794654od_a_b @ Y3 @ Ys ) ) )
           => ( P @ Xs2 @ Ys3 ) ) ) ) ) ).

% list_induct2'
thf(fact_638_list__induct2_H,axiom,
    ! [P: list_a > list_P1467974523212632366od_a_b > $o,Xs2: list_a,Ys3: list_P1467974523212632366od_a_b] :
      ( ( P @ nil_a @ nil_Product_prod_a_b )
     => ( ! [X3: a,Xs: list_a] : ( P @ ( cons_a @ X3 @ Xs ) @ nil_Product_prod_a_b )
       => ( ! [Y3: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] : ( P @ nil_a @ ( cons_P7316939131009794654od_a_b @ Y3 @ Ys ) )
         => ( ! [X3: a,Xs: list_a,Y3: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_a @ X3 @ Xs ) @ ( cons_P7316939131009794654od_a_b @ Y3 @ Ys ) ) )
           => ( P @ Xs2 @ Ys3 ) ) ) ) ) ).

% list_induct2'
thf(fact_639_list__induct2_H,axiom,
    ! [P: list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > $o,Xs2: list_P1467974523212632366od_a_b,Ys3: list_P1467974523212632366od_a_b] :
      ( ( P @ nil_Product_prod_a_b @ nil_Product_prod_a_b )
     => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] : ( P @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ nil_Product_prod_a_b )
       => ( ! [Y3: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] : ( P @ nil_Product_prod_a_b @ ( cons_P7316939131009794654od_a_b @ Y3 @ Ys ) )
         => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Y3: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ ( cons_P7316939131009794654od_a_b @ Y3 @ Ys ) ) )
           => ( P @ Xs2 @ Ys3 ) ) ) ) ) ).

% list_induct2'
thf(fact_640_list__induct2_H,axiom,
    ! [P: list_m9210641325446214647_a_b_a > list_b > $o,Xs2: list_m9210641325446214647_a_b_a,Ys3: list_b] :
      ( ( P @ nil_ma546712388551201761_a_b_a @ nil_b )
     => ( ! [X3: mappin2365780136655028071_a_b_a,Xs: list_m9210641325446214647_a_b_a] : ( P @ ( cons_m8574943908289518641_a_b_a @ X3 @ Xs ) @ nil_b )
       => ( ! [Y3: b,Ys: list_b] : ( P @ nil_ma546712388551201761_a_b_a @ ( cons_b @ Y3 @ Ys ) )
         => ( ! [X3: mappin2365780136655028071_a_b_a,Xs: list_m9210641325446214647_a_b_a,Y3: b,Ys: list_b] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_m8574943908289518641_a_b_a @ X3 @ Xs ) @ ( cons_b @ Y3 @ Ys ) ) )
           => ( P @ Xs2 @ Ys3 ) ) ) ) ) ).

% list_induct2'
thf(fact_641_list__nonempty__induct,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,P: list_P1467974523212632366od_a_b > $o] :
      ( ( Xs2 != nil_Product_prod_a_b )
     => ( ! [X3: product_prod_a_b] : ( P @ ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) )
       => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
              ( ( Xs != nil_Product_prod_a_b )
             => ( ( P @ Xs )
               => ( P @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% list_nonempty_induct
thf(fact_642_list__nonempty__induct,axiom,
    ! [Xs2: list_m9210641325446214647_a_b_a,P: list_m9210641325446214647_a_b_a > $o] :
      ( ( Xs2 != nil_ma546712388551201761_a_b_a )
     => ( ! [X3: mappin2365780136655028071_a_b_a] : ( P @ ( cons_m8574943908289518641_a_b_a @ X3 @ nil_ma546712388551201761_a_b_a ) )
       => ( ! [X3: mappin2365780136655028071_a_b_a,Xs: list_m9210641325446214647_a_b_a] :
              ( ( Xs != nil_ma546712388551201761_a_b_a )
             => ( ( P @ Xs )
               => ( P @ ( cons_m8574943908289518641_a_b_a @ X3 @ Xs ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% list_nonempty_induct
thf(fact_643_list__nonempty__induct,axiom,
    ! [Xs2: list_l337834041258660428_d_nat,P: list_l337834041258660428_d_nat > $o] :
      ( ( Xs2 != nil_li1008261075667615414_d_nat )
     => ( ! [X3: list_P6684558232516339260_d_nat] : ( P @ ( cons_l4410424432949683974_d_nat @ X3 @ nil_li1008261075667615414_d_nat ) )
       => ( ! [X3: list_P6684558232516339260_d_nat,Xs: list_l337834041258660428_d_nat] :
              ( ( Xs != nil_li1008261075667615414_d_nat )
             => ( ( P @ Xs )
               => ( P @ ( cons_l4410424432949683974_d_nat @ X3 @ Xs ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% list_nonempty_induct
thf(fact_644_list__nonempty__induct,axiom,
    ! [Xs2: list_b,P: list_b > $o] :
      ( ( Xs2 != nil_b )
     => ( ! [X3: b] : ( P @ ( cons_b @ X3 @ nil_b ) )
       => ( ! [X3: b,Xs: list_b] :
              ( ( Xs != nil_b )
             => ( ( P @ Xs )
               => ( P @ ( cons_b @ X3 @ Xs ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% list_nonempty_induct
thf(fact_645_list__nonempty__induct,axiom,
    ! [Xs2: list_a,P: list_a > $o] :
      ( ( Xs2 != nil_a )
     => ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
       => ( ! [X3: a,Xs: list_a] :
              ( ( Xs != nil_a )
             => ( ( P @ Xs )
               => ( P @ ( cons_a @ X3 @ Xs ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% list_nonempty_induct
thf(fact_646_foldl__Cons,axiom,
    ! [F: produc2062766750192416900_a_b_a > produc1990351791824763820_d_nat > produc2062766750192416900_a_b_a,A: produc2062766750192416900_a_b_a,X: produc1990351791824763820_d_nat,Xs2: list_P6684558232516339260_d_nat] :
      ( ( foldl_3722756957450160079_d_nat @ F @ A @ ( cons_P2594832900118292854_d_nat @ X @ Xs2 ) )
      = ( foldl_3722756957450160079_d_nat @ F @ ( F @ A @ X ) @ Xs2 ) ) ).

% foldl_Cons
thf(fact_647_foldl__Cons,axiom,
    ! [F: list_P6684558232516339260_d_nat > produc1990351791824763820_d_nat > list_P6684558232516339260_d_nat,A: list_P6684558232516339260_d_nat,X: produc1990351791824763820_d_nat,Xs2: list_P6684558232516339260_d_nat] :
      ( ( foldl_5754872400397474315_d_nat @ F @ A @ ( cons_P2594832900118292854_d_nat @ X @ Xs2 ) )
      = ( foldl_5754872400397474315_d_nat @ F @ ( F @ A @ X ) @ Xs2 ) ) ).

% foldl_Cons
thf(fact_648_lookup__map__entry__neq,axiom,
    ! [X: produc7822699449245016212_a_b_a,Y5: produc7822699449245016212_a_b_a,F: a > a,M: mappin4996136453398522101_b_a_a] :
      ( ( X != Y5 )
     => ( ( lookup1409243802988840364_b_a_a @ ( map_en8183204059568693808_b_a_a @ X @ F @ M ) @ Y5 )
        = ( lookup1409243802988840364_b_a_a @ M @ Y5 ) ) ) ).

% lookup_map_entry_neq
thf(fact_649_lookup__map__entry__neq,axiom,
    ! [X: list_P6684558232516339260_d_nat,Y5: list_P6684558232516339260_d_nat,F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( X != Y5 )
     => ( ( lookup4661578797684577249_a_b_a @ ( map_en7372027836849589853_a_b_a @ X @ F @ M ) @ Y5 )
        = ( lookup4661578797684577249_a_b_a @ M @ Y5 ) ) ) ).

% lookup_map_entry_neq
thf(fact_650_lookup__map__entry__neq,axiom,
    ! [X: a,Y5: a,F: produc5456268773499131775_d_nat > produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( X != Y5 )
     => ( ( lookup8480750236328838337_d_nat @ ( map_en1595895783147578181_d_nat @ X @ F @ M ) @ Y5 )
        = ( lookup8480750236328838337_d_nat @ M @ Y5 ) ) ) ).

% lookup_map_entry_neq
thf(fact_651_lookup__map__entry__neq,axiom,
    ! [X: a,Y5: a,F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( X != Y5 )
     => ( ( lookup1261341372242361769_a_b_a @ ( map_en8511718966582312493_a_b_a @ X @ F @ M ) @ Y5 )
        = ( lookup1261341372242361769_a_b_a @ M @ Y5 ) ) ) ).

% lookup_map_entry_neq
thf(fact_652_lookup__map__entry__neq,axiom,
    ! [X: a,Y5: a,F: b > b,M: mapping_a_b] :
      ( ( X != Y5 )
     => ( ( lookup_a_b @ ( map_entry_a_b @ X @ F @ M ) @ Y5 )
        = ( lookup_a_b @ M @ Y5 ) ) ) ).

% lookup_map_entry_neq
thf(fact_653_lookup__map__entry__neq,axiom,
    ! [X: product_prod_a_b,Y5: product_prod_a_b,F: a > a,M: mappin2365780136655028071_a_b_a] :
      ( ( X != Y5 )
     => ( ( lookup863726980359885586_a_b_a @ ( map_en8135975850489418638_a_b_a @ X @ F @ M ) @ Y5 )
        = ( lookup863726980359885586_a_b_a @ M @ Y5 ) ) ) ).

% lookup_map_entry_neq
thf(fact_654_mmap__combine_Osimps_I2_J,axiom,
    ! [K2: a,V2: mappin4996136453398522101_b_a_a,C2: mappin4996136453398522101_b_a_a > mappin4996136453398522101_b_a_a > mappin4996136453398522101_b_a_a,P2: produc3619439354540788382_b_a_a,Ps: list_P3594502398639369124_b_a_a] :
      ( ( mmap_c6646943743142367051_b_a_a @ K2 @ V2 @ C2 @ ( cons_P220927179099053908_b_a_a @ P2 @ Ps ) )
      = ( produc8712858204422192577_b_a_a
        @ ^ [K4: a,V3: mappin4996136453398522101_b_a_a] : ( if_lis2561630526582691434_b_a_a @ ( K2 = K4 ) @ ( cons_P220927179099053908_b_a_a @ ( produc4259011374121840526_b_a_a @ K2 @ ( C2 @ V3 @ V2 ) ) @ Ps ) @ ( cons_P220927179099053908_b_a_a @ P2 @ ( mmap_c6646943743142367051_b_a_a @ K2 @ V2 @ C2 @ Ps ) ) )
        @ P2 ) ) ).

% mmap_combine.simps(2)
thf(fact_655_mmap__combine_Osimps_I2_J,axiom,
    ! [K2: a,V2: list_P6684558232516339260_d_nat,C2: list_P6684558232516339260_d_nat > list_P6684558232516339260_d_nat > list_P6684558232516339260_d_nat,P2: produc9176665399011442921_d_nat,Ps: list_P9063253934932097401_d_nat] :
      ( ( mmap_c2696983453546349414_d_nat @ K2 @ V2 @ C2 @ ( cons_P260773292763473843_d_nat @ P2 @ Ps ) )
      = ( produc4823120784119823497_d_nat
        @ ^ [K4: a,V3: list_P6684558232516339260_d_nat] : ( if_lis3214374446479449651_d_nat @ ( K2 = K4 ) @ ( cons_P260773292763473843_d_nat @ ( produc919746265332163299_d_nat @ K2 @ ( C2 @ V3 @ V2 ) ) @ Ps ) @ ( cons_P260773292763473843_d_nat @ P2 @ ( mmap_c2696983453546349414_d_nat @ K2 @ V2 @ C2 @ Ps ) ) )
        @ P2 ) ) ).

% mmap_combine.simps(2)
thf(fact_656_mmap__combine_Osimps_I2_J,axiom,
    ! [K2: a,V2: list_P2992919015626629540_a_b_a,C2: list_P2992919015626629540_a_b_a > list_P2992919015626629540_a_b_a > list_P2992919015626629540_a_b_a,P2: produc7083967959564351825_a_b_a,Ps: list_P3172577687508813921_a_b_a] :
      ( ( mmap_c13429169768989646_a_b_a @ K2 @ V2 @ C2 @ ( cons_P2675685553303450907_a_b_a @ P2 @ Ps ) )
      = ( produc2435392120593834073_a_b_a
        @ ^ [K4: a,V3: list_P2992919015626629540_a_b_a] : ( if_lis3166759124437449243_a_b_a @ ( K2 = K4 ) @ ( cons_P2675685553303450907_a_b_a @ ( produc6994869176599099851_a_b_a @ K2 @ ( C2 @ V3 @ V2 ) ) @ Ps ) @ ( cons_P2675685553303450907_a_b_a @ P2 @ ( mmap_c13429169768989646_a_b_a @ K2 @ V2 @ C2 @ Ps ) ) )
        @ P2 ) ) ).

% mmap_combine.simps(2)
thf(fact_657_mmap__combine_Osimps_I2_J,axiom,
    ! [K2: a,V2: list_P1467974523212632366od_a_b,C2: list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b,P2: produc2639161875728825943od_a_b,Ps: list_P5700792096213675741od_a_b] :
      ( ( mmap_c5556936107559350148od_a_b @ K2 @ V2 @ C2 @ ( cons_P8483812562729421581od_a_b @ P2 @ Ps ) )
      = ( produc3480138538282384385od_a_b
        @ ^ [K4: a,V3: list_P1467974523212632366od_a_b] : ( if_lis8754265742650888611od_a_b @ ( K2 = K4 ) @ ( cons_P8483812562729421581od_a_b @ ( produc6643264353547226055od_a_b @ K2 @ ( C2 @ V3 @ V2 ) ) @ Ps ) @ ( cons_P8483812562729421581od_a_b @ P2 @ ( mmap_c5556936107559350148od_a_b @ K2 @ V2 @ C2 @ Ps ) ) )
        @ P2 ) ) ).

% mmap_combine.simps(2)
thf(fact_658_mmap__combine_Osimps_I2_J,axiom,
    ! [K2: a,V2: a,C2: a > a > a,P2: product_prod_a_a,Ps: list_P1396940483166286381od_a_a] :
      ( ( mmap_combine_a_a @ K2 @ V2 @ C2 @ ( cons_P7316939126706565853od_a_a @ P2 @ Ps ) )
      = ( produc6057368760105205761od_a_a
        @ ^ [K4: a,V3: a] : ( if_lis931442767461590515od_a_a @ ( K2 = K4 ) @ ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ K2 @ ( C2 @ V3 @ V2 ) ) @ Ps ) @ ( cons_P7316939126706565853od_a_a @ P2 @ ( mmap_combine_a_a @ K2 @ V2 @ C2 @ Ps ) ) )
        @ P2 ) ) ).

% mmap_combine.simps(2)
thf(fact_659_mmap__combine_Osimps_I2_J,axiom,
    ! [K2: a,V2: produc5456268773499131775_d_nat,C2: produc5456268773499131775_d_nat > produc5456268773499131775_d_nat > produc5456268773499131775_d_nat,P2: produc1990351791824763820_d_nat,Ps: list_P6684558232516339260_d_nat] :
      ( ( mmap_c2205071097579735977_d_nat @ K2 @ V2 @ C2 @ ( cons_P2594832900118292854_d_nat @ P2 @ Ps ) )
      = ( produc4938868912396122127_d_nat
        @ ^ [K4: a,V3: produc5456268773499131775_d_nat] : ( if_lis7731207453108189686_d_nat @ ( K2 = K4 ) @ ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K2 @ ( C2 @ V3 @ V2 ) ) @ Ps ) @ ( cons_P2594832900118292854_d_nat @ P2 @ ( mmap_c2205071097579735977_d_nat @ K2 @ V2 @ C2 @ Ps ) ) )
        @ P2 ) ) ).

% mmap_combine.simps(2)
thf(fact_660_mmap__combine_Osimps_I2_J,axiom,
    ! [K2: a,V2: b,C2: b > b > b,P2: product_prod_a_b,Ps: list_P1467974523212632366od_a_b] :
      ( ( mmap_combine_a_b @ K2 @ V2 @ C2 @ ( cons_P7316939131009794654od_a_b @ P2 @ Ps ) )
      = ( produc611504906096598721od_a_b
        @ ^ [K4: a,V3: b] : ( if_lis1002476807507936500od_a_b @ ( K2 = K4 ) @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ ( C2 @ V3 @ V2 ) ) @ Ps ) @ ( cons_P7316939131009794654od_a_b @ P2 @ ( mmap_combine_a_b @ K2 @ V2 @ C2 @ Ps ) ) )
        @ P2 ) ) ).

% mmap_combine.simps(2)
thf(fact_661_mmap__combine_Osimps_I2_J,axiom,
    ! [K2: produc1990351791824763820_d_nat,V2: mappin2365780136655028071_a_b_a,C2: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,P2: produc6343861673719254644_a_b_a,Ps: list_P4793483941206178810_a_b_a] :
      ( ( mmap_c5580072897124408041_a_b_a @ K2 @ V2 @ C2 @ ( cons_P673726611873665578_a_b_a @ P2 @ Ps ) )
      = ( produc7463616454677732345_a_b_a
        @ ^ [K4: produc1990351791824763820_d_nat,V3: mappin2365780136655028071_a_b_a] : ( if_lis8426428089627639232_a_b_a @ ( K2 = K4 ) @ ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K2 @ ( C2 @ V3 @ V2 ) ) @ Ps ) @ ( cons_P673726611873665578_a_b_a @ P2 @ ( mmap_c5580072897124408041_a_b_a @ K2 @ V2 @ C2 @ Ps ) ) )
        @ P2 ) ) ).

% mmap_combine.simps(2)
thf(fact_662_mmap__combine_Osimps_I2_J,axiom,
    ! [K2: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a,C2: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,P2: produc2062766750192416900_a_b_a,Ps: list_P7611574810425128074_a_b_a] :
      ( ( mmap_c8213366964327184121_a_b_a @ K2 @ V2 @ C2 @ ( cons_P2804281896780507450_a_b_a @ P2 @ Ps ) )
      = ( produc2202866172573321337_a_b_a
        @ ^ [K4: list_P6684558232516339260_d_nat,V3: mappin2365780136655028071_a_b_a] : ( if_lis6818758761248955216_a_b_a @ ( K2 = K4 ) @ ( cons_P2804281896780507450_a_b_a @ ( produc2616710729536263484_a_b_a @ K2 @ ( C2 @ V3 @ V2 ) ) @ Ps ) @ ( cons_P2804281896780507450_a_b_a @ P2 @ ( mmap_c8213366964327184121_a_b_a @ K2 @ V2 @ C2 @ Ps ) ) )
        @ P2 ) ) ).

% mmap_combine.simps(2)
thf(fact_663_mmap__combine_Osimps_I2_J,axiom,
    ! [K2: a,V2: mappin2365780136655028071_a_b_a,C2: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,P2: produc7822699449245016212_a_b_a,Ps: list_P2992919015626629540_a_b_a] :
      ( ( mmap_c402980803363502225_a_b_a @ K2 @ V2 @ C2 @ ( cons_P5314874490607035230_a_b_a @ P2 @ Ps ) )
      = ( produc4579919614062918239_a_b_a
        @ ^ [K4: a,V3: mappin2365780136655028071_a_b_a] : ( if_lis7797837208463966814_a_b_a @ ( K2 = K4 ) @ ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K2 @ ( C2 @ V3 @ V2 ) ) @ Ps ) @ ( cons_P5314874490607035230_a_b_a @ P2 @ ( mmap_c402980803363502225_a_b_a @ K2 @ V2 @ C2 @ Ps ) ) )
        @ P2 ) ) ).

% mmap_combine.simps(2)
thf(fact_664_delete__aux_Ocases,axiom,
    ! [X: produc4743123391170388173_b_a_a] :
      ( ! [K5: a] :
          ( X
         != ( produc4204554423659385021_b_a_a @ K5 @ nil_Pr5740214961507490724_b_a_a ) )
     => ~ ! [K5: a,K6: a,V4: mappin4996136453398522101_b_a_a,Xs: list_P3594502398639369124_b_a_a] :
            ( X
           != ( produc4204554423659385021_b_a_a @ K5 @ ( cons_P220927179099053908_b_a_a @ ( produc4259011374121840526_b_a_a @ K6 @ V4 ) @ Xs ) ) ) ) ).

% delete_aux.cases
thf(fact_665_delete__aux_Ocases,axiom,
    ! [X: produc5968402719899699878_d_nat] :
      ( ! [K5: a] :
          ( X
         != ( produc5205599450977154976_d_nat @ K5 @ nil_Pr1577584438861778787_d_nat ) )
     => ~ ! [K5: a,K6: a,V4: list_P6684558232516339260_d_nat,Xs: list_P9063253934932097401_d_nat] :
            ( X
           != ( produc5205599450977154976_d_nat @ K5 @ ( cons_P260773292763473843_d_nat @ ( produc919746265332163299_d_nat @ K6 @ V4 ) @ Xs ) ) ) ) ).

% delete_aux.cases
thf(fact_666_delete__aux_Ocases,axiom,
    ! [X: produc3755538318669010574_a_b_a] :
      ( ! [K5: a] :
          ( X
         != ( produc1039823637343887880_a_b_a @ K5 @ nil_Pr7283556224815412427_a_b_a ) )
     => ~ ! [K5: a,K6: a,V4: list_P2992919015626629540_a_b_a,Xs: list_P3172577687508813921_a_b_a] :
            ( X
           != ( produc1039823637343887880_a_b_a @ K5 @ ( cons_P2675685553303450907_a_b_a @ ( produc6994869176599099851_a_b_a @ K6 @ V4 ) @ Xs ) ) ) ) ).

% delete_aux.cases
thf(fact_667_delete__aux_Ocases,axiom,
    ! [X: produc5068203645811208070od_a_b] :
      ( ! [K5: a] :
          ( X
         != ( produc7441261679724955638od_a_b @ K5 @ nil_Pr7243433834293086045od_a_b ) )
     => ~ ! [K5: a,K6: a,V4: list_P1467974523212632366od_a_b,Xs: list_P5700792096213675741od_a_b] :
            ( X
           != ( produc7441261679724955638od_a_b @ K5 @ ( cons_P8483812562729421581od_a_b @ ( produc6643264353547226055od_a_b @ K6 @ V4 ) @ Xs ) ) ) ) ).

% delete_aux.cases
thf(fact_668_delete__aux_Ocases,axiom,
    ! [X: produc5160346394017861590od_a_a] :
      ( ! [K5: a] :
          ( X
         != ( produc6572230313500880070od_a_a @ K5 @ nil_Product_prod_a_a ) )
     => ~ ! [K5: a,K6: a,V4: a,Xs: list_P1396940483166286381od_a_a] :
            ( X
           != ( produc6572230313500880070od_a_a @ K5 @ ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ K6 @ V4 ) @ Xs ) ) ) ) ).

% delete_aux.cases
thf(fact_669_delete__aux_Ocases,axiom,
    ! [X: produc9176665399011442921_d_nat] :
      ( ! [K5: a] :
          ( X
         != ( produc919746265332163299_d_nat @ K5 @ nil_Pr2300798825815202086_d_nat ) )
     => ~ ! [K5: a,K6: a,V4: produc5456268773499131775_d_nat,Xs: list_P6684558232516339260_d_nat] :
            ( X
           != ( produc919746265332163299_d_nat @ K5 @ ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K6 @ V4 ) @ Xs ) ) ) ) ).

% delete_aux.cases
thf(fact_670_delete__aux_Ocases,axiom,
    ! [X: produc2639161875728825943od_a_b] :
      ( ! [K5: a] :
          ( X
         != ( produc6643264353547226055od_a_b @ K5 @ nil_Product_prod_a_b ) )
     => ~ ! [K5: a,K6: a,V4: b,Xs: list_P1467974523212632366od_a_b] :
            ( X
           != ( produc6643264353547226055od_a_b @ K5 @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K6 @ V4 ) @ Xs ) ) ) ) ).

% delete_aux.cases
thf(fact_671_delete__aux_Ocases,axiom,
    ! [X: produc101150017066992451_a_b_a] :
      ( ! [K5: produc1990351791824763820_d_nat] :
          ( X
         != ( produc712968421101327221_a_b_a @ K5 @ nil_Pr3763312433997567610_a_b_a ) )
     => ~ ! [K5: produc1990351791824763820_d_nat,K6: produc1990351791824763820_d_nat,V4: mappin2365780136655028071_a_b_a,Xs: list_P4793483941206178810_a_b_a] :
            ( X
           != ( produc712968421101327221_a_b_a @ K5 @ ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K6 @ V4 ) @ Xs ) ) ) ) ).

% delete_aux.cases
thf(fact_672_delete__aux_Ocases,axiom,
    ! [X: produc6655655590374074051_a_b_a] :
      ( ! [K5: list_P6684558232516339260_d_nat] :
          ( X
         != ( produc8862864803594885109_a_b_a @ K5 @ nil_Pr1908111780246848906_a_b_a ) )
     => ~ ! [K5: list_P6684558232516339260_d_nat,K6: list_P6684558232516339260_d_nat,V4: mappin2365780136655028071_a_b_a,Xs: list_P7611574810425128074_a_b_a] :
            ( X
           != ( produc8862864803594885109_a_b_a @ K5 @ ( cons_P2804281896780507450_a_b_a @ ( produc2616710729536263484_a_b_a @ K6 @ V4 ) @ Xs ) ) ) ) ).

% delete_aux.cases
thf(fact_673_delete__aux_Ocases,axiom,
    ! [X: produc7083967959564351825_a_b_a] :
      ( ! [K5: a] :
          ( X
         != ( produc6994869176599099851_a_b_a @ K5 @ nil_Pr7632893056509733646_a_b_a ) )
     => ~ ! [K5: a,K6: a,V4: mappin2365780136655028071_a_b_a,Xs: list_P2992919015626629540_a_b_a] :
            ( X
           != ( produc6994869176599099851_a_b_a @ K5 @ ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K6 @ V4 ) @ Xs ) ) ) ) ).

% delete_aux.cases
thf(fact_674_map__entry_Ocases,axiom,
    ! [X: produc2913117349713790443od_a_b] :
      ( ! [K5: a,F4: b > b] :
          ( X
         != ( produc928590970379683685od_a_b @ K5 @ ( produc1507266648589652720od_a_b @ F4 @ nil_Product_prod_a_b ) ) )
     => ~ ! [K5: a,F4: b > b,P3: product_prod_a_b,Ps2: list_P1467974523212632366od_a_b] :
            ( X
           != ( produc928590970379683685od_a_b @ K5 @ ( produc1507266648589652720od_a_b @ F4 @ ( cons_P7316939131009794654od_a_b @ P3 @ Ps2 ) ) ) ) ) ).

% map_entry.cases
thf(fact_675_map__default_Ocases,axiom,
    ! [X: produc5344594549881938713od_a_b] :
      ( ! [K5: a,V4: b,F4: b > b] :
          ( X
         != ( produc3839279985106316947od_a_b @ K5 @ ( produc6726914776229624550od_a_b @ V4 @ ( produc1507266648589652720od_a_b @ F4 @ nil_Product_prod_a_b ) ) ) )
     => ~ ! [K5: a,V4: b,F4: b > b,P3: product_prod_a_b,Ps2: list_P1467974523212632366od_a_b] :
            ( X
           != ( produc3839279985106316947od_a_b @ K5 @ ( produc6726914776229624550od_a_b @ V4 @ ( produc1507266648589652720od_a_b @ F4 @ ( cons_P7316939131009794654od_a_b @ P3 @ Ps2 ) ) ) ) ) ) ).

% map_default.cases
thf(fact_676_clearjunk_Ocases,axiom,
    ! [X: list_P1467974523212632366od_a_b] :
      ( ( X != nil_Product_prod_a_b )
     => ~ ! [P3: product_prod_a_b,Ps2: list_P1467974523212632366od_a_b] :
            ( X
           != ( cons_P7316939131009794654od_a_b @ P3 @ Ps2 ) ) ) ).

% clearjunk.cases
thf(fact_677_insert__Nil,axiom,
    ! [X: product_prod_a_b] :
      ( ( insert7736115125267272132od_a_b @ X @ nil_Product_prod_a_b )
      = ( cons_P7316939131009794654od_a_b @ X @ nil_Product_prod_a_b ) ) ).

% insert_Nil
thf(fact_678_insert__Nil,axiom,
    ! [X: mappin2365780136655028071_a_b_a] :
      ( ( insert2343651869219193675_a_b_a @ X @ nil_ma546712388551201761_a_b_a )
      = ( cons_m8574943908289518641_a_b_a @ X @ nil_ma546712388551201761_a_b_a ) ) ).

% insert_Nil
thf(fact_679_insert__Nil,axiom,
    ! [X: list_P6684558232516339260_d_nat] :
      ( ( insert2293668203985506848_d_nat @ X @ nil_li1008261075667615414_d_nat )
      = ( cons_l4410424432949683974_d_nat @ X @ nil_li1008261075667615414_d_nat ) ) ).

% insert_Nil
thf(fact_680_insert__Nil,axiom,
    ! [X: b] :
      ( ( insert_b @ X @ nil_b )
      = ( cons_b @ X @ nil_b ) ) ).

% insert_Nil
thf(fact_681_insert__Nil,axiom,
    ! [X: a] :
      ( ( insert_a @ X @ nil_a )
      = ( cons_a @ X @ nil_a ) ) ).

% insert_Nil
thf(fact_682_option_Oinject,axiom,
    ! [X2: produc7822699449245016212_a_b_a,Y2: produc7822699449245016212_a_b_a] :
      ( ( ( some_P2288122801350569071_a_b_a @ X2 )
        = ( some_P2288122801350569071_a_b_a @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_683_option_Oinject,axiom,
    ! [X2: product_prod_a_b,Y2: product_prod_a_b] :
      ( ( ( some_P3592067299498605709od_a_b @ X2 )
        = ( some_P3592067299498605709od_a_b @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_684_option_Oinject,axiom,
    ! [X2: mappin2365780136655028071_a_b_a,Y2: mappin2365780136655028071_a_b_a] :
      ( ( ( some_m5820982296856387138_a_b_a @ X2 )
        = ( some_m5820982296856387138_a_b_a @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_685_option_Oinject,axiom,
    ! [X2: b,Y2: b] :
      ( ( ( some_b @ X2 )
        = ( some_b @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_686_option_Oinject,axiom,
    ! [X2: a,Y2: a] :
      ( ( ( some_a @ X2 )
        = ( some_a @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_687_lookup__update,axiom,
    ! [K2: a,V2: produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( lookup8480750236328838337_d_nat @ ( update6116250764560905298_d_nat @ K2 @ V2 @ M ) @ K2 )
      = ( some_P3314015041498521818_d_nat @ V2 ) ) ).

% lookup_update
thf(fact_688_lookup__update,axiom,
    ! [K2: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( lookup4661578797684577249_a_b_a @ ( update5108779570945132176_a_b_a @ K2 @ V2 @ M ) @ K2 )
      = ( some_m5820982296856387138_a_b_a @ V2 ) ) ).

% lookup_update
thf(fact_689_lookup__update,axiom,
    ! [K2: a,V2: mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( lookup1261341372242361769_a_b_a @ ( update5632971445410033082_a_b_a @ K2 @ V2 @ M ) @ K2 )
      = ( some_m5820982296856387138_a_b_a @ V2 ) ) ).

% lookup_update
thf(fact_690_lookup__update,axiom,
    ! [K2: a,V2: b,M: mapping_a_b] :
      ( ( lookup_a_b @ ( update_a_b @ K2 @ V2 @ M ) @ K2 )
      = ( some_b @ V2 ) ) ).

% lookup_update
thf(fact_691_lookup__update,axiom,
    ! [K2: produc7822699449245016212_a_b_a,V2: a,M: mappin4996136453398522101_b_a_a] :
      ( ( lookup1409243802988840364_b_a_a @ ( update5685058447907601597_b_a_a @ K2 @ V2 @ M ) @ K2 )
      = ( some_a @ V2 ) ) ).

% lookup_update
thf(fact_692_lookup__update,axiom,
    ! [K2: product_prod_a_b,V2: a,M: mappin2365780136655028071_a_b_a] :
      ( ( lookup863726980359885586_a_b_a @ ( update751575294834384705_a_b_a @ K2 @ V2 @ M ) @ K2 )
      = ( some_a @ V2 ) ) ).

% lookup_update
thf(fact_693_option_Osimps_I5_J,axiom,
    ! [F1: produc7822699449245016212_a_b_a,F22: a > produc7822699449245016212_a_b_a,X2: a] :
      ( ( case_o140846598303662877_b_a_a @ F1 @ F22 @ ( some_a @ X2 ) )
      = ( F22 @ X2 ) ) ).

% option.simps(5)
thf(fact_694_option_Osimps_I5_J,axiom,
    ! [F1: mappin2365780136655028071_a_b_a,F22: a > mappin2365780136655028071_a_b_a,X2: a] :
      ( ( case_o2154183385986904842_b_a_a @ F1 @ F22 @ ( some_a @ X2 ) )
      = ( F22 @ X2 ) ) ).

% option.simps(5)
thf(fact_695_option_Osimps_I5_J,axiom,
    ! [F1: option3893003681250311888_d_nat,F22: product_prod_d_nat > option3893003681250311888_d_nat,X2: product_prod_d_nat] :
      ( ( case_o6579539980594861021_d_nat @ F1 @ F22 @ ( some_P734314052486210395_d_nat @ X2 ) )
      = ( F22 @ X2 ) ) ).

% option.simps(5)
thf(fact_696_option_Osimps_I5_J,axiom,
    ! [F1: option_a,F22: a > option_a,X2: a] :
      ( ( case_o3148979394504432965on_a_a @ F1 @ F22 @ ( some_a @ X2 ) )
      = ( F22 @ X2 ) ) ).

% option.simps(5)
thf(fact_697_option_Osimps_I5_J,axiom,
    ! [F1: a,F22: a > a,X2: a] :
      ( ( case_option_a_a @ F1 @ F22 @ ( some_a @ X2 ) )
      = ( F22 @ X2 ) ) ).

% option.simps(5)
thf(fact_698_option_Osimps_I5_J,axiom,
    ! [F1: $o,F22: a > $o,X2: a] :
      ( ( case_option_o_a @ F1 @ F22 @ ( some_a @ X2 ) )
      = ( F22 @ X2 ) ) ).

% option.simps(5)
thf(fact_699_lookup__update_H,axiom,
    ! [K2: a,K3: a,V2: produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( ( K2 = K3 )
       => ( ( lookup8480750236328838337_d_nat @ ( update6116250764560905298_d_nat @ K2 @ V2 @ M ) @ K3 )
          = ( some_P3314015041498521818_d_nat @ V2 ) ) )
      & ( ( K2 != K3 )
       => ( ( lookup8480750236328838337_d_nat @ ( update6116250764560905298_d_nat @ K2 @ V2 @ M ) @ K3 )
          = ( lookup8480750236328838337_d_nat @ M @ K3 ) ) ) ) ).

% lookup_update'
thf(fact_700_lookup__update_H,axiom,
    ! [K2: list_P6684558232516339260_d_nat,K3: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( ( K2 = K3 )
       => ( ( lookup4661578797684577249_a_b_a @ ( update5108779570945132176_a_b_a @ K2 @ V2 @ M ) @ K3 )
          = ( some_m5820982296856387138_a_b_a @ V2 ) ) )
      & ( ( K2 != K3 )
       => ( ( lookup4661578797684577249_a_b_a @ ( update5108779570945132176_a_b_a @ K2 @ V2 @ M ) @ K3 )
          = ( lookup4661578797684577249_a_b_a @ M @ K3 ) ) ) ) ).

% lookup_update'
thf(fact_701_lookup__update_H,axiom,
    ! [K2: a,K3: a,V2: mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( ( K2 = K3 )
       => ( ( lookup1261341372242361769_a_b_a @ ( update5632971445410033082_a_b_a @ K2 @ V2 @ M ) @ K3 )
          = ( some_m5820982296856387138_a_b_a @ V2 ) ) )
      & ( ( K2 != K3 )
       => ( ( lookup1261341372242361769_a_b_a @ ( update5632971445410033082_a_b_a @ K2 @ V2 @ M ) @ K3 )
          = ( lookup1261341372242361769_a_b_a @ M @ K3 ) ) ) ) ).

% lookup_update'
thf(fact_702_lookup__update_H,axiom,
    ! [K2: a,K3: a,V2: b,M: mapping_a_b] :
      ( ( ( K2 = K3 )
       => ( ( lookup_a_b @ ( update_a_b @ K2 @ V2 @ M ) @ K3 )
          = ( some_b @ V2 ) ) )
      & ( ( K2 != K3 )
       => ( ( lookup_a_b @ ( update_a_b @ K2 @ V2 @ M ) @ K3 )
          = ( lookup_a_b @ M @ K3 ) ) ) ) ).

% lookup_update'
thf(fact_703_lookup__update_H,axiom,
    ! [K2: produc7822699449245016212_a_b_a,K3: produc7822699449245016212_a_b_a,V2: a,M: mappin4996136453398522101_b_a_a] :
      ( ( ( K2 = K3 )
       => ( ( lookup1409243802988840364_b_a_a @ ( update5685058447907601597_b_a_a @ K2 @ V2 @ M ) @ K3 )
          = ( some_a @ V2 ) ) )
      & ( ( K2 != K3 )
       => ( ( lookup1409243802988840364_b_a_a @ ( update5685058447907601597_b_a_a @ K2 @ V2 @ M ) @ K3 )
          = ( lookup1409243802988840364_b_a_a @ M @ K3 ) ) ) ) ).

% lookup_update'
thf(fact_704_lookup__update_H,axiom,
    ! [K2: product_prod_a_b,K3: product_prod_a_b,V2: a,M: mappin2365780136655028071_a_b_a] :
      ( ( ( K2 = K3 )
       => ( ( lookup863726980359885586_a_b_a @ ( update751575294834384705_a_b_a @ K2 @ V2 @ M ) @ K3 )
          = ( some_a @ V2 ) ) )
      & ( ( K2 != K3 )
       => ( ( lookup863726980359885586_a_b_a @ ( update751575294834384705_a_b_a @ K2 @ V2 @ M ) @ K3 )
          = ( lookup863726980359885586_a_b_a @ M @ K3 ) ) ) ) ).

% lookup_update'
thf(fact_705_lookup__default,axiom,
    ! [X: a,D2: produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( lookup8480750236328838337_d_nat @ ( defaul4873115978101149310_d_nat @ X @ D2 @ M ) @ X )
      = ( some_P3314015041498521818_d_nat @ ( lookup2689911013986210168_nat_a @ D2 @ M @ X ) ) ) ).

% lookup_default
thf(fact_706_lookup__default,axiom,
    ! [X: list_P6684558232516339260_d_nat,D2: mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( lookup4661578797684577249_a_b_a @ ( defaul2939598897860386660_a_b_a @ X @ D2 @ M ) @ X )
      = ( some_m5820982296856387138_a_b_a @ ( lookup1777753518785040938_d_nat @ D2 @ M @ X ) ) ) ).

% lookup_default
thf(fact_707_lookup__default,axiom,
    ! [X: a,D2: mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( lookup1261341372242361769_a_b_a @ ( defaul1545336500834713062_a_b_a @ X @ D2 @ M ) @ X )
      = ( some_m5820982296856387138_a_b_a @ ( lookup4260590398720651408_b_a_a @ D2 @ M @ X ) ) ) ).

% lookup_default
thf(fact_708_lookup__default,axiom,
    ! [X: a,D2: b,M: mapping_a_b] :
      ( ( lookup_a_b @ ( default_a_b @ X @ D2 @ M ) @ X )
      = ( some_b @ ( lookup_default_b_a @ D2 @ M @ X ) ) ) ).

% lookup_default
thf(fact_709_lookup__default,axiom,
    ! [X: produc7822699449245016212_a_b_a,D2: a,M: mappin4996136453398522101_b_a_a] :
      ( ( lookup1409243802988840364_b_a_a @ ( defaul2000508409559777513_b_a_a @ X @ D2 @ M ) @ X )
      = ( some_a @ ( lookup4806362731325673933_a_b_a @ D2 @ M @ X ) ) ) ).

% lookup_default
thf(fact_710_lookup__default,axiom,
    ! [X: product_prod_a_b,D2: a,M: mappin2365780136655028071_a_b_a] :
      ( ( lookup863726980359885586_a_b_a @ ( defaul4527334963600521749_a_b_a @ X @ D2 @ M ) @ X )
      = ( some_a @ ( lookup592102128023449775od_a_b @ D2 @ M @ X ) ) ) ).

% lookup_default
thf(fact_711_lookup__default_H,axiom,
    ! [X: a,Y5: a,D2: produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( ( X = Y5 )
       => ( ( lookup8480750236328838337_d_nat @ ( defaul4873115978101149310_d_nat @ X @ D2 @ M ) @ Y5 )
          = ( some_P3314015041498521818_d_nat @ ( lookup2689911013986210168_nat_a @ D2 @ M @ X ) ) ) )
      & ( ( X != Y5 )
       => ( ( lookup8480750236328838337_d_nat @ ( defaul4873115978101149310_d_nat @ X @ D2 @ M ) @ Y5 )
          = ( lookup8480750236328838337_d_nat @ M @ Y5 ) ) ) ) ).

% lookup_default'
thf(fact_712_lookup__default_H,axiom,
    ! [X: list_P6684558232516339260_d_nat,Y5: list_P6684558232516339260_d_nat,D2: mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( ( X = Y5 )
       => ( ( lookup4661578797684577249_a_b_a @ ( defaul2939598897860386660_a_b_a @ X @ D2 @ M ) @ Y5 )
          = ( some_m5820982296856387138_a_b_a @ ( lookup1777753518785040938_d_nat @ D2 @ M @ X ) ) ) )
      & ( ( X != Y5 )
       => ( ( lookup4661578797684577249_a_b_a @ ( defaul2939598897860386660_a_b_a @ X @ D2 @ M ) @ Y5 )
          = ( lookup4661578797684577249_a_b_a @ M @ Y5 ) ) ) ) ).

% lookup_default'
thf(fact_713_lookup__default_H,axiom,
    ! [X: a,Y5: a,D2: mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( ( X = Y5 )
       => ( ( lookup1261341372242361769_a_b_a @ ( defaul1545336500834713062_a_b_a @ X @ D2 @ M ) @ Y5 )
          = ( some_m5820982296856387138_a_b_a @ ( lookup4260590398720651408_b_a_a @ D2 @ M @ X ) ) ) )
      & ( ( X != Y5 )
       => ( ( lookup1261341372242361769_a_b_a @ ( defaul1545336500834713062_a_b_a @ X @ D2 @ M ) @ Y5 )
          = ( lookup1261341372242361769_a_b_a @ M @ Y5 ) ) ) ) ).

% lookup_default'
thf(fact_714_lookup__default_H,axiom,
    ! [X: a,Y5: a,D2: b,M: mapping_a_b] :
      ( ( ( X = Y5 )
       => ( ( lookup_a_b @ ( default_a_b @ X @ D2 @ M ) @ Y5 )
          = ( some_b @ ( lookup_default_b_a @ D2 @ M @ X ) ) ) )
      & ( ( X != Y5 )
       => ( ( lookup_a_b @ ( default_a_b @ X @ D2 @ M ) @ Y5 )
          = ( lookup_a_b @ M @ Y5 ) ) ) ) ).

% lookup_default'
thf(fact_715_lookup__default_H,axiom,
    ! [X: produc7822699449245016212_a_b_a,Y5: produc7822699449245016212_a_b_a,D2: a,M: mappin4996136453398522101_b_a_a] :
      ( ( ( X = Y5 )
       => ( ( lookup1409243802988840364_b_a_a @ ( defaul2000508409559777513_b_a_a @ X @ D2 @ M ) @ Y5 )
          = ( some_a @ ( lookup4806362731325673933_a_b_a @ D2 @ M @ X ) ) ) )
      & ( ( X != Y5 )
       => ( ( lookup1409243802988840364_b_a_a @ ( defaul2000508409559777513_b_a_a @ X @ D2 @ M ) @ Y5 )
          = ( lookup1409243802988840364_b_a_a @ M @ Y5 ) ) ) ) ).

% lookup_default'
thf(fact_716_lookup__default_H,axiom,
    ! [X: product_prod_a_b,Y5: product_prod_a_b,D2: a,M: mappin2365780136655028071_a_b_a] :
      ( ( ( X = Y5 )
       => ( ( lookup863726980359885586_a_b_a @ ( defaul4527334963600521749_a_b_a @ X @ D2 @ M ) @ Y5 )
          = ( some_a @ ( lookup592102128023449775od_a_b @ D2 @ M @ X ) ) ) )
      & ( ( X != Y5 )
       => ( ( lookup863726980359885586_a_b_a @ ( defaul4527334963600521749_a_b_a @ X @ D2 @ M ) @ Y5 )
          = ( lookup863726980359885586_a_b_a @ M @ Y5 ) ) ) ) ).

% lookup_default'
thf(fact_717_lookup__map__default,axiom,
    ! [X: a,D2: produc5456268773499131775_d_nat,F: produc5456268773499131775_d_nat > produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( lookup8480750236328838337_d_nat @ ( map_de521311723871066358_d_nat @ X @ D2 @ F @ M ) @ X )
      = ( some_P3314015041498521818_d_nat @ ( F @ ( lookup2689911013986210168_nat_a @ D2 @ M @ X ) ) ) ) ).

% lookup_map_default
thf(fact_718_lookup__map__default,axiom,
    ! [X: list_P6684558232516339260_d_nat,D2: mappin2365780136655028071_a_b_a,F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( lookup4661578797684577249_a_b_a @ ( map_de582901244306572012_a_b_a @ X @ D2 @ F @ M ) @ X )
      = ( some_m5820982296856387138_a_b_a @ ( F @ ( lookup1777753518785040938_d_nat @ D2 @ M @ X ) ) ) ) ).

% lookup_map_default
thf(fact_719_lookup__map__default,axiom,
    ! [X: a,D2: mappin2365780136655028071_a_b_a,F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( lookup1261341372242361769_a_b_a @ ( map_de1440600275304793182_a_b_a @ X @ D2 @ F @ M ) @ X )
      = ( some_m5820982296856387138_a_b_a @ ( F @ ( lookup4260590398720651408_b_a_a @ D2 @ M @ X ) ) ) ) ).

% lookup_map_default
thf(fact_720_lookup__map__default,axiom,
    ! [X: a,D2: b,F: b > b,M: mapping_a_b] :
      ( ( lookup_a_b @ ( map_default_a_b2 @ X @ D2 @ F @ M ) @ X )
      = ( some_b @ ( F @ ( lookup_default_b_a @ D2 @ M @ X ) ) ) ) ).

% lookup_map_default
thf(fact_721_lookup__map__default,axiom,
    ! [X: produc7822699449245016212_a_b_a,D2: a,F: a > a,M: mappin4996136453398522101_b_a_a] :
      ( ( lookup1409243802988840364_b_a_a @ ( map_de3308694818481590113_b_a_a @ X @ D2 @ F @ M ) @ X )
      = ( some_a @ ( F @ ( lookup4806362731325673933_a_b_a @ D2 @ M @ X ) ) ) ) ).

% lookup_map_default
thf(fact_722_lookup__map__default,axiom,
    ! [X: product_prod_a_b,D2: a,F: a > a,M: mappin2365780136655028071_a_b_a] :
      ( ( lookup863726980359885586_a_b_a @ ( map_de4839670203443761565_a_b_a @ X @ D2 @ F @ M ) @ X )
      = ( some_a @ ( F @ ( lookup592102128023449775od_a_b @ D2 @ M @ X ) ) ) ) ).

% lookup_map_default
thf(fact_723_lookup__map__default_H,axiom,
    ! [X: a,Y5: a,D2: produc5456268773499131775_d_nat,F: produc5456268773499131775_d_nat > produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( ( X = Y5 )
       => ( ( lookup8480750236328838337_d_nat @ ( map_de521311723871066358_d_nat @ X @ D2 @ F @ M ) @ Y5 )
          = ( some_P3314015041498521818_d_nat @ ( F @ ( lookup2689911013986210168_nat_a @ D2 @ M @ X ) ) ) ) )
      & ( ( X != Y5 )
       => ( ( lookup8480750236328838337_d_nat @ ( map_de521311723871066358_d_nat @ X @ D2 @ F @ M ) @ Y5 )
          = ( lookup8480750236328838337_d_nat @ M @ Y5 ) ) ) ) ).

% lookup_map_default'
thf(fact_724_lookup__map__default_H,axiom,
    ! [X: list_P6684558232516339260_d_nat,Y5: list_P6684558232516339260_d_nat,D2: mappin2365780136655028071_a_b_a,F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( ( X = Y5 )
       => ( ( lookup4661578797684577249_a_b_a @ ( map_de582901244306572012_a_b_a @ X @ D2 @ F @ M ) @ Y5 )
          = ( some_m5820982296856387138_a_b_a @ ( F @ ( lookup1777753518785040938_d_nat @ D2 @ M @ X ) ) ) ) )
      & ( ( X != Y5 )
       => ( ( lookup4661578797684577249_a_b_a @ ( map_de582901244306572012_a_b_a @ X @ D2 @ F @ M ) @ Y5 )
          = ( lookup4661578797684577249_a_b_a @ M @ Y5 ) ) ) ) ).

% lookup_map_default'
thf(fact_725_lookup__map__default_H,axiom,
    ! [X: a,Y5: a,D2: mappin2365780136655028071_a_b_a,F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( ( X = Y5 )
       => ( ( lookup1261341372242361769_a_b_a @ ( map_de1440600275304793182_a_b_a @ X @ D2 @ F @ M ) @ Y5 )
          = ( some_m5820982296856387138_a_b_a @ ( F @ ( lookup4260590398720651408_b_a_a @ D2 @ M @ X ) ) ) ) )
      & ( ( X != Y5 )
       => ( ( lookup1261341372242361769_a_b_a @ ( map_de1440600275304793182_a_b_a @ X @ D2 @ F @ M ) @ Y5 )
          = ( lookup1261341372242361769_a_b_a @ M @ Y5 ) ) ) ) ).

% lookup_map_default'
thf(fact_726_lookup__map__default_H,axiom,
    ! [X: a,Y5: a,D2: b,F: b > b,M: mapping_a_b] :
      ( ( ( X = Y5 )
       => ( ( lookup_a_b @ ( map_default_a_b2 @ X @ D2 @ F @ M ) @ Y5 )
          = ( some_b @ ( F @ ( lookup_default_b_a @ D2 @ M @ X ) ) ) ) )
      & ( ( X != Y5 )
       => ( ( lookup_a_b @ ( map_default_a_b2 @ X @ D2 @ F @ M ) @ Y5 )
          = ( lookup_a_b @ M @ Y5 ) ) ) ) ).

% lookup_map_default'
thf(fact_727_lookup__map__default_H,axiom,
    ! [X: produc7822699449245016212_a_b_a,Y5: produc7822699449245016212_a_b_a,D2: a,F: a > a,M: mappin4996136453398522101_b_a_a] :
      ( ( ( X = Y5 )
       => ( ( lookup1409243802988840364_b_a_a @ ( map_de3308694818481590113_b_a_a @ X @ D2 @ F @ M ) @ Y5 )
          = ( some_a @ ( F @ ( lookup4806362731325673933_a_b_a @ D2 @ M @ X ) ) ) ) )
      & ( ( X != Y5 )
       => ( ( lookup1409243802988840364_b_a_a @ ( map_de3308694818481590113_b_a_a @ X @ D2 @ F @ M ) @ Y5 )
          = ( lookup1409243802988840364_b_a_a @ M @ Y5 ) ) ) ) ).

% lookup_map_default'
thf(fact_728_lookup__map__default_H,axiom,
    ! [X: product_prod_a_b,Y5: product_prod_a_b,D2: a,F: a > a,M: mappin2365780136655028071_a_b_a] :
      ( ( ( X = Y5 )
       => ( ( lookup863726980359885586_a_b_a @ ( map_de4839670203443761565_a_b_a @ X @ D2 @ F @ M ) @ Y5 )
          = ( some_a @ ( F @ ( lookup592102128023449775od_a_b @ D2 @ M @ X ) ) ) ) )
      & ( ( X != Y5 )
       => ( ( lookup863726980359885586_a_b_a @ ( map_de4839670203443761565_a_b_a @ X @ D2 @ F @ M ) @ Y5 )
          = ( lookup863726980359885586_a_b_a @ M @ Y5 ) ) ) ) ).

% lookup_map_default'
thf(fact_729_product__lists_Osimps_I1_J,axiom,
    ( ( produc5891818806577749399od_a_b @ nil_li8635777946366267444od_a_b )
    = ( cons_l907386227682280420od_a_b @ nil_Product_prod_a_b @ nil_li8635777946366267444od_a_b ) ) ).

% product_lists.simps(1)
thf(fact_730_product__lists_Osimps_I1_J,axiom,
    ( ( produc115545289656543928_a_b_a @ nil_li7899906211823649649_a_b_a )
    = ( cons_l8438923613303137217_a_b_a @ nil_ma546712388551201761_a_b_a @ nil_li7899906211823649649_a_b_a ) ) ).

% product_lists.simps(1)
thf(fact_731_product__lists_Osimps_I1_J,axiom,
    ( ( produc6448671210887846797_d_nat @ nil_li4515601735571940806_d_nat )
    = ( cons_l8566576933256865302_d_nat @ nil_li1008261075667615414_d_nat @ nil_li4515601735571940806_d_nat ) ) ).

% product_lists.simps(1)
thf(fact_732_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_b @ nil_list_b )
    = ( cons_list_b @ nil_b @ nil_list_b ) ) ).

% product_lists.simps(1)
thf(fact_733_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_a @ nil_list_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% product_lists.simps(1)
thf(fact_734_product__lists_Osimps_I1_J,axiom,
    ( ( produc6617437221750312317_d_nat @ nil_li1008261075667615414_d_nat )
    = ( cons_l4410424432949683974_d_nat @ nil_Pr2300798825815202086_d_nat @ nil_li1008261075667615414_d_nat ) ) ).

% product_lists.simps(1)
thf(fact_735_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: mappin4996136453398522101_b_a_a,K2: a,F: mappin4996136453398522101_b_a_a > mappin4996136453398522101_b_a_a] :
      ( ( update7444102524143346635_a_a_a @ V2 @ K2 @ F @ nil_Pr5740214961507490724_b_a_a )
      = ( cons_P220927179099053908_b_a_a @ ( produc4259011374121840526_b_a_a @ K2 @ ( F @ V2 ) ) @ nil_Pr5740214961507490724_b_a_a ) ) ).

% update_with_aux.simps(1)
thf(fact_736_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: list_P6684558232516339260_d_nat,K2: a,F: list_P6684558232516339260_d_nat > list_P6684558232516339260_d_nat] :
      ( ( update6061945500663741502_nat_a @ V2 @ K2 @ F @ nil_Pr1577584438861778787_d_nat )
      = ( cons_P260773292763473843_d_nat @ ( produc919746265332163299_d_nat @ K2 @ ( F @ V2 ) ) @ nil_Pr1577584438861778787_d_nat ) ) ).

% update_with_aux.simps(1)
thf(fact_737_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: list_P2992919015626629540_a_b_a,K2: a,F: list_P2992919015626629540_a_b_a > list_P2992919015626629540_a_b_a] :
      ( ( update6636503624010838358_b_a_a @ V2 @ K2 @ F @ nil_Pr7283556224815412427_a_b_a )
      = ( cons_P2675685553303450907_a_b_a @ ( produc6994869176599099851_a_b_a @ K2 @ ( F @ V2 ) ) @ nil_Pr7283556224815412427_a_b_a ) ) ).

% update_with_aux.simps(1)
thf(fact_738_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: list_P1467974523212632366od_a_b,K2: a,F: list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b] :
      ( ( update5571748460587972562_a_b_a @ V2 @ K2 @ F @ nil_Pr7243433834293086045od_a_b )
      = ( cons_P8483812562729421581od_a_b @ ( produc6643264353547226055od_a_b @ K2 @ ( F @ V2 ) ) @ nil_Pr7243433834293086045od_a_b ) ) ).

% update_with_aux.simps(1)
thf(fact_739_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: a,K2: a,F: a > a] :
      ( ( update_with_aux_a_a @ V2 @ K2 @ F @ nil_Product_prod_a_a )
      = ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ K2 @ ( F @ V2 ) ) @ nil_Product_prod_a_a ) ) ).

% update_with_aux.simps(1)
thf(fact_740_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: produc5456268773499131775_d_nat,K2: a,F: produc5456268773499131775_d_nat > produc5456268773499131775_d_nat] :
      ( ( update3832787365488499003_nat_a @ V2 @ K2 @ F @ nil_Pr2300798825815202086_d_nat )
      = ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K2 @ ( F @ V2 ) ) @ nil_Pr2300798825815202086_d_nat ) ) ).

% update_with_aux.simps(1)
thf(fact_741_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: b,K2: a,F: b > b] :
      ( ( update_with_aux_b_a @ V2 @ K2 @ F @ nil_Product_prod_a_b )
      = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ ( F @ V2 ) ) @ nil_Product_prod_a_b ) ) ).

% update_with_aux.simps(1)
thf(fact_742_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: mappin2365780136655028071_a_b_a,K2: produc1990351791824763820_d_nat,F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a] :
      ( ( update6341529825412160535_d_nat @ V2 @ K2 @ F @ nil_Pr3763312433997567610_a_b_a )
      = ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K2 @ ( F @ V2 ) ) @ nil_Pr3763312433997567610_a_b_a ) ) ).

% update_with_aux.simps(1)
thf(fact_743_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: mappin2365780136655028071_a_b_a,K2: list_P6684558232516339260_d_nat,F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a] :
      ( ( update2279199214462932263_d_nat @ V2 @ K2 @ F @ nil_Pr1908111780246848906_a_b_a )
      = ( cons_P2804281896780507450_a_b_a @ ( produc2616710729536263484_a_b_a @ K2 @ ( F @ V2 ) ) @ nil_Pr1908111780246848906_a_b_a ) ) ).

% update_with_aux.simps(1)
thf(fact_744_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: mappin2365780136655028071_a_b_a,K2: a,F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a] :
      ( ( update5782414861844070867_b_a_a @ V2 @ K2 @ F @ nil_Pr7632893056509733646_a_b_a )
      = ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K2 @ ( F @ V2 ) ) @ nil_Pr7632893056509733646_a_b_a ) ) ).

% update_with_aux.simps(1)
thf(fact_745_subseqs_Osimps_I1_J,axiom,
    ( ( subseq3159716953272533757od_a_b @ nil_Product_prod_a_b )
    = ( cons_l907386227682280420od_a_b @ nil_Product_prod_a_b @ nil_li8635777946366267444od_a_b ) ) ).

% subseqs.simps(1)
thf(fact_746_subseqs_Osimps_I1_J,axiom,
    ( ( subseq9190561028072782802_a_b_a @ nil_ma546712388551201761_a_b_a )
    = ( cons_l8438923613303137217_a_b_a @ nil_ma546712388551201761_a_b_a @ nil_li7899906211823649649_a_b_a ) ) ).

% subseqs.simps(1)
thf(fact_747_subseqs_Osimps_I1_J,axiom,
    ( ( subseq344413824759591079_d_nat @ nil_li1008261075667615414_d_nat )
    = ( cons_l8566576933256865302_d_nat @ nil_li1008261075667615414_d_nat @ nil_li4515601735571940806_d_nat ) ) ).

% subseqs.simps(1)
thf(fact_748_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_b @ nil_b )
    = ( cons_list_b @ nil_b @ nil_list_b ) ) ).

% subseqs.simps(1)
thf(fact_749_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_a @ nil_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% subseqs.simps(1)
thf(fact_750_subseqs_Osimps_I1_J,axiom,
    ( ( subseq6476932015355475863_d_nat @ nil_Pr2300798825815202086_d_nat )
    = ( cons_l4410424432949683974_d_nat @ nil_Pr2300798825815202086_d_nat @ nil_li1008261075667615414_d_nat ) ) ).

% subseqs.simps(1)
thf(fact_751_accp__induct__rule,axiom,
    ! [R3: produc7083967959564351825_a_b_a > produc7083967959564351825_a_b_a > $o,A: produc7083967959564351825_a_b_a,P: produc7083967959564351825_a_b_a > $o] :
      ( ( accp_P209492061542291080_a_b_a @ R3 @ A )
     => ( ! [X3: produc7083967959564351825_a_b_a] :
            ( ( accp_P209492061542291080_a_b_a @ R3 @ X3 )
           => ( ! [Y8: produc7083967959564351825_a_b_a] :
                  ( ( R3 @ Y8 @ X3 )
                 => ( P @ Y8 ) )
             => ( P @ X3 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_752_accp__induct__rule,axiom,
    ! [R3: produc6655655590374074051_a_b_a > produc6655655590374074051_a_b_a > $o,A: produc6655655590374074051_a_b_a,P: produc6655655590374074051_a_b_a > $o] :
      ( ( accp_P6506438267451010426_a_b_a @ R3 @ A )
     => ( ! [X3: produc6655655590374074051_a_b_a] :
            ( ( accp_P6506438267451010426_a_b_a @ R3 @ X3 )
           => ( ! [Y8: produc6655655590374074051_a_b_a] :
                  ( ( R3 @ Y8 @ X3 )
                 => ( P @ Y8 ) )
             => ( P @ X3 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_753_accp__induct__rule,axiom,
    ! [R3: produc101150017066992451_a_b_a > produc101150017066992451_a_b_a > $o,A: produc101150017066992451_a_b_a,P: produc101150017066992451_a_b_a > $o] :
      ( ( accp_P8454841393138718970_a_b_a @ R3 @ A )
     => ( ! [X3: produc101150017066992451_a_b_a] :
            ( ( accp_P8454841393138718970_a_b_a @ R3 @ X3 )
           => ( ! [Y8: produc101150017066992451_a_b_a] :
                  ( ( R3 @ Y8 @ X3 )
                 => ( P @ Y8 ) )
             => ( P @ X3 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_754_accp__induct__rule,axiom,
    ! [R3: produc2639161875728825943od_a_b > produc2639161875728825943od_a_b > $o,A: produc2639161875728825943od_a_b,P: produc2639161875728825943od_a_b > $o] :
      ( ( accp_P6845111368451614688od_a_b @ R3 @ A )
     => ( ! [X3: produc2639161875728825943od_a_b] :
            ( ( accp_P6845111368451614688od_a_b @ R3 @ X3 )
           => ( ! [Y8: produc2639161875728825943od_a_b] :
                  ( ( R3 @ Y8 @ X3 )
                 => ( P @ Y8 ) )
             => ( P @ X3 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_755_accp__induct__rule,axiom,
    ! [R3: produc9176665399011442921_d_nat > produc9176665399011442921_d_nat > $o,A: produc9176665399011442921_d_nat,P: produc9176665399011442921_d_nat > $o] :
      ( ( accp_P7714024643000875680_d_nat @ R3 @ A )
     => ( ! [X3: produc9176665399011442921_d_nat] :
            ( ( accp_P7714024643000875680_d_nat @ R3 @ X3 )
           => ( ! [Y8: produc9176665399011442921_d_nat] :
                  ( ( R3 @ Y8 @ X3 )
                 => ( P @ Y8 ) )
             => ( P @ X3 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_756_not__accp__down,axiom,
    ! [R: produc7083967959564351825_a_b_a > produc7083967959564351825_a_b_a > $o,X: produc7083967959564351825_a_b_a] :
      ( ~ ( accp_P209492061542291080_a_b_a @ R @ X )
     => ~ ! [Z4: produc7083967959564351825_a_b_a] :
            ( ( R @ Z4 @ X )
           => ( accp_P209492061542291080_a_b_a @ R @ Z4 ) ) ) ).

% not_accp_down
thf(fact_757_not__accp__down,axiom,
    ! [R: produc6655655590374074051_a_b_a > produc6655655590374074051_a_b_a > $o,X: produc6655655590374074051_a_b_a] :
      ( ~ ( accp_P6506438267451010426_a_b_a @ R @ X )
     => ~ ! [Z4: produc6655655590374074051_a_b_a] :
            ( ( R @ Z4 @ X )
           => ( accp_P6506438267451010426_a_b_a @ R @ Z4 ) ) ) ).

% not_accp_down
thf(fact_758_not__accp__down,axiom,
    ! [R: produc101150017066992451_a_b_a > produc101150017066992451_a_b_a > $o,X: produc101150017066992451_a_b_a] :
      ( ~ ( accp_P8454841393138718970_a_b_a @ R @ X )
     => ~ ! [Z4: produc101150017066992451_a_b_a] :
            ( ( R @ Z4 @ X )
           => ( accp_P8454841393138718970_a_b_a @ R @ Z4 ) ) ) ).

% not_accp_down
thf(fact_759_not__accp__down,axiom,
    ! [R: produc2639161875728825943od_a_b > produc2639161875728825943od_a_b > $o,X: produc2639161875728825943od_a_b] :
      ( ~ ( accp_P6845111368451614688od_a_b @ R @ X )
     => ~ ! [Z4: produc2639161875728825943od_a_b] :
            ( ( R @ Z4 @ X )
           => ( accp_P6845111368451614688od_a_b @ R @ Z4 ) ) ) ).

% not_accp_down
thf(fact_760_not__accp__down,axiom,
    ! [R: produc9176665399011442921_d_nat > produc9176665399011442921_d_nat > $o,X: produc9176665399011442921_d_nat] :
      ( ~ ( accp_P7714024643000875680_d_nat @ R @ X )
     => ~ ! [Z4: produc9176665399011442921_d_nat] :
            ( ( R @ Z4 @ X )
           => ( accp_P7714024643000875680_d_nat @ R @ Z4 ) ) ) ).

% not_accp_down
thf(fact_761_accp__downward,axiom,
    ! [R3: produc7083967959564351825_a_b_a > produc7083967959564351825_a_b_a > $o,B: produc7083967959564351825_a_b_a,A: produc7083967959564351825_a_b_a] :
      ( ( accp_P209492061542291080_a_b_a @ R3 @ B )
     => ( ( R3 @ A @ B )
       => ( accp_P209492061542291080_a_b_a @ R3 @ A ) ) ) ).

% accp_downward
thf(fact_762_accp__downward,axiom,
    ! [R3: produc6655655590374074051_a_b_a > produc6655655590374074051_a_b_a > $o,B: produc6655655590374074051_a_b_a,A: produc6655655590374074051_a_b_a] :
      ( ( accp_P6506438267451010426_a_b_a @ R3 @ B )
     => ( ( R3 @ A @ B )
       => ( accp_P6506438267451010426_a_b_a @ R3 @ A ) ) ) ).

% accp_downward
thf(fact_763_accp__downward,axiom,
    ! [R3: produc101150017066992451_a_b_a > produc101150017066992451_a_b_a > $o,B: produc101150017066992451_a_b_a,A: produc101150017066992451_a_b_a] :
      ( ( accp_P8454841393138718970_a_b_a @ R3 @ B )
     => ( ( R3 @ A @ B )
       => ( accp_P8454841393138718970_a_b_a @ R3 @ A ) ) ) ).

% accp_downward
thf(fact_764_accp__downward,axiom,
    ! [R3: produc2639161875728825943od_a_b > produc2639161875728825943od_a_b > $o,B: produc2639161875728825943od_a_b,A: produc2639161875728825943od_a_b] :
      ( ( accp_P6845111368451614688od_a_b @ R3 @ B )
     => ( ( R3 @ A @ B )
       => ( accp_P6845111368451614688od_a_b @ R3 @ A ) ) ) ).

% accp_downward
thf(fact_765_accp__downward,axiom,
    ! [R3: produc9176665399011442921_d_nat > produc9176665399011442921_d_nat > $o,B: produc9176665399011442921_d_nat,A: produc9176665399011442921_d_nat] :
      ( ( accp_P7714024643000875680_d_nat @ R3 @ B )
     => ( ( R3 @ A @ B )
       => ( accp_P7714024643000875680_d_nat @ R3 @ A ) ) ) ).

% accp_downward
thf(fact_766_accp__induct,axiom,
    ! [R3: produc7083967959564351825_a_b_a > produc7083967959564351825_a_b_a > $o,A: produc7083967959564351825_a_b_a,P: produc7083967959564351825_a_b_a > $o] :
      ( ( accp_P209492061542291080_a_b_a @ R3 @ A )
     => ( ! [X3: produc7083967959564351825_a_b_a] :
            ( ( accp_P209492061542291080_a_b_a @ R3 @ X3 )
           => ( ! [Y8: produc7083967959564351825_a_b_a] :
                  ( ( R3 @ Y8 @ X3 )
                 => ( P @ Y8 ) )
             => ( P @ X3 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_767_accp__induct,axiom,
    ! [R3: produc6655655590374074051_a_b_a > produc6655655590374074051_a_b_a > $o,A: produc6655655590374074051_a_b_a,P: produc6655655590374074051_a_b_a > $o] :
      ( ( accp_P6506438267451010426_a_b_a @ R3 @ A )
     => ( ! [X3: produc6655655590374074051_a_b_a] :
            ( ( accp_P6506438267451010426_a_b_a @ R3 @ X3 )
           => ( ! [Y8: produc6655655590374074051_a_b_a] :
                  ( ( R3 @ Y8 @ X3 )
                 => ( P @ Y8 ) )
             => ( P @ X3 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_768_accp__induct,axiom,
    ! [R3: produc101150017066992451_a_b_a > produc101150017066992451_a_b_a > $o,A: produc101150017066992451_a_b_a,P: produc101150017066992451_a_b_a > $o] :
      ( ( accp_P8454841393138718970_a_b_a @ R3 @ A )
     => ( ! [X3: produc101150017066992451_a_b_a] :
            ( ( accp_P8454841393138718970_a_b_a @ R3 @ X3 )
           => ( ! [Y8: produc101150017066992451_a_b_a] :
                  ( ( R3 @ Y8 @ X3 )
                 => ( P @ Y8 ) )
             => ( P @ X3 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_769_accp__induct,axiom,
    ! [R3: produc2639161875728825943od_a_b > produc2639161875728825943od_a_b > $o,A: produc2639161875728825943od_a_b,P: produc2639161875728825943od_a_b > $o] :
      ( ( accp_P6845111368451614688od_a_b @ R3 @ A )
     => ( ! [X3: produc2639161875728825943od_a_b] :
            ( ( accp_P6845111368451614688od_a_b @ R3 @ X3 )
           => ( ! [Y8: produc2639161875728825943od_a_b] :
                  ( ( R3 @ Y8 @ X3 )
                 => ( P @ Y8 ) )
             => ( P @ X3 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_770_accp__induct,axiom,
    ! [R3: produc9176665399011442921_d_nat > produc9176665399011442921_d_nat > $o,A: produc9176665399011442921_d_nat,P: produc9176665399011442921_d_nat > $o] :
      ( ( accp_P7714024643000875680_d_nat @ R3 @ A )
     => ( ! [X3: produc9176665399011442921_d_nat] :
            ( ( accp_P7714024643000875680_d_nat @ R3 @ X3 )
           => ( ! [Y8: produc9176665399011442921_d_nat] :
                  ( ( R3 @ Y8 @ X3 )
                 => ( P @ Y8 ) )
             => ( P @ X3 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_771_accpI,axiom,
    ! [R3: produc7083967959564351825_a_b_a > produc7083967959564351825_a_b_a > $o,X: produc7083967959564351825_a_b_a] :
      ( ! [Y3: produc7083967959564351825_a_b_a] :
          ( ( R3 @ Y3 @ X )
         => ( accp_P209492061542291080_a_b_a @ R3 @ Y3 ) )
     => ( accp_P209492061542291080_a_b_a @ R3 @ X ) ) ).

% accpI
thf(fact_772_accpI,axiom,
    ! [R3: produc6655655590374074051_a_b_a > produc6655655590374074051_a_b_a > $o,X: produc6655655590374074051_a_b_a] :
      ( ! [Y3: produc6655655590374074051_a_b_a] :
          ( ( R3 @ Y3 @ X )
         => ( accp_P6506438267451010426_a_b_a @ R3 @ Y3 ) )
     => ( accp_P6506438267451010426_a_b_a @ R3 @ X ) ) ).

% accpI
thf(fact_773_accpI,axiom,
    ! [R3: produc101150017066992451_a_b_a > produc101150017066992451_a_b_a > $o,X: produc101150017066992451_a_b_a] :
      ( ! [Y3: produc101150017066992451_a_b_a] :
          ( ( R3 @ Y3 @ X )
         => ( accp_P8454841393138718970_a_b_a @ R3 @ Y3 ) )
     => ( accp_P8454841393138718970_a_b_a @ R3 @ X ) ) ).

% accpI
thf(fact_774_accpI,axiom,
    ! [R3: produc2639161875728825943od_a_b > produc2639161875728825943od_a_b > $o,X: produc2639161875728825943od_a_b] :
      ( ! [Y3: produc2639161875728825943od_a_b] :
          ( ( R3 @ Y3 @ X )
         => ( accp_P6845111368451614688od_a_b @ R3 @ Y3 ) )
     => ( accp_P6845111368451614688od_a_b @ R3 @ X ) ) ).

% accpI
thf(fact_775_accpI,axiom,
    ! [R3: produc9176665399011442921_d_nat > produc9176665399011442921_d_nat > $o,X: produc9176665399011442921_d_nat] :
      ( ! [Y3: produc9176665399011442921_d_nat] :
          ( ( R3 @ Y3 @ X )
         => ( accp_P7714024643000875680_d_nat @ R3 @ Y3 ) )
     => ( accp_P7714024643000875680_d_nat @ R3 @ X ) ) ).

% accpI
thf(fact_776_accp_Osimps,axiom,
    ( accp_P209492061542291080_a_b_a
    = ( ^ [R2: produc7083967959564351825_a_b_a > produc7083967959564351825_a_b_a > $o,A5: produc7083967959564351825_a_b_a] :
        ? [X4: produc7083967959564351825_a_b_a] :
          ( ( A5 = X4 )
          & ! [Y4: produc7083967959564351825_a_b_a] :
              ( ( R2 @ Y4 @ X4 )
             => ( accp_P209492061542291080_a_b_a @ R2 @ Y4 ) ) ) ) ) ).

% accp.simps
thf(fact_777_accp_Osimps,axiom,
    ( accp_P6506438267451010426_a_b_a
    = ( ^ [R2: produc6655655590374074051_a_b_a > produc6655655590374074051_a_b_a > $o,A5: produc6655655590374074051_a_b_a] :
        ? [X4: produc6655655590374074051_a_b_a] :
          ( ( A5 = X4 )
          & ! [Y4: produc6655655590374074051_a_b_a] :
              ( ( R2 @ Y4 @ X4 )
             => ( accp_P6506438267451010426_a_b_a @ R2 @ Y4 ) ) ) ) ) ).

% accp.simps
thf(fact_778_accp_Osimps,axiom,
    ( accp_P8454841393138718970_a_b_a
    = ( ^ [R2: produc101150017066992451_a_b_a > produc101150017066992451_a_b_a > $o,A5: produc101150017066992451_a_b_a] :
        ? [X4: produc101150017066992451_a_b_a] :
          ( ( A5 = X4 )
          & ! [Y4: produc101150017066992451_a_b_a] :
              ( ( R2 @ Y4 @ X4 )
             => ( accp_P8454841393138718970_a_b_a @ R2 @ Y4 ) ) ) ) ) ).

% accp.simps
thf(fact_779_accp_Osimps,axiom,
    ( accp_P6845111368451614688od_a_b
    = ( ^ [R2: produc2639161875728825943od_a_b > produc2639161875728825943od_a_b > $o,A5: produc2639161875728825943od_a_b] :
        ? [X4: produc2639161875728825943od_a_b] :
          ( ( A5 = X4 )
          & ! [Y4: produc2639161875728825943od_a_b] :
              ( ( R2 @ Y4 @ X4 )
             => ( accp_P6845111368451614688od_a_b @ R2 @ Y4 ) ) ) ) ) ).

% accp.simps
thf(fact_780_accp_Osimps,axiom,
    ( accp_P7714024643000875680_d_nat
    = ( ^ [R2: produc9176665399011442921_d_nat > produc9176665399011442921_d_nat > $o,A5: produc9176665399011442921_d_nat] :
        ? [X4: produc9176665399011442921_d_nat] :
          ( ( A5 = X4 )
          & ! [Y4: produc9176665399011442921_d_nat] :
              ( ( R2 @ Y4 @ X4 )
             => ( accp_P7714024643000875680_d_nat @ R2 @ Y4 ) ) ) ) ) ).

% accp.simps
thf(fact_781_accp_Ocases,axiom,
    ! [R3: produc7083967959564351825_a_b_a > produc7083967959564351825_a_b_a > $o,A: produc7083967959564351825_a_b_a] :
      ( ( accp_P209492061542291080_a_b_a @ R3 @ A )
     => ! [Y8: produc7083967959564351825_a_b_a] :
          ( ( R3 @ Y8 @ A )
         => ( accp_P209492061542291080_a_b_a @ R3 @ Y8 ) ) ) ).

% accp.cases
thf(fact_782_accp_Ocases,axiom,
    ! [R3: produc6655655590374074051_a_b_a > produc6655655590374074051_a_b_a > $o,A: produc6655655590374074051_a_b_a] :
      ( ( accp_P6506438267451010426_a_b_a @ R3 @ A )
     => ! [Y8: produc6655655590374074051_a_b_a] :
          ( ( R3 @ Y8 @ A )
         => ( accp_P6506438267451010426_a_b_a @ R3 @ Y8 ) ) ) ).

% accp.cases
thf(fact_783_accp_Ocases,axiom,
    ! [R3: produc101150017066992451_a_b_a > produc101150017066992451_a_b_a > $o,A: produc101150017066992451_a_b_a] :
      ( ( accp_P8454841393138718970_a_b_a @ R3 @ A )
     => ! [Y8: produc101150017066992451_a_b_a] :
          ( ( R3 @ Y8 @ A )
         => ( accp_P8454841393138718970_a_b_a @ R3 @ Y8 ) ) ) ).

% accp.cases
thf(fact_784_accp_Ocases,axiom,
    ! [R3: produc2639161875728825943od_a_b > produc2639161875728825943od_a_b > $o,A: produc2639161875728825943od_a_b] :
      ( ( accp_P6845111368451614688od_a_b @ R3 @ A )
     => ! [Y8: produc2639161875728825943od_a_b] :
          ( ( R3 @ Y8 @ A )
         => ( accp_P6845111368451614688od_a_b @ R3 @ Y8 ) ) ) ).

% accp.cases
thf(fact_785_accp_Ocases,axiom,
    ! [R3: produc9176665399011442921_d_nat > produc9176665399011442921_d_nat > $o,A: produc9176665399011442921_d_nat] :
      ( ( accp_P7714024643000875680_d_nat @ R3 @ A )
     => ! [Y8: produc9176665399011442921_d_nat] :
          ( ( R3 @ Y8 @ A )
         => ( accp_P7714024643000875680_d_nat @ R3 @ Y8 ) ) ) ).

% accp.cases
thf(fact_786_map__default__def,axiom,
    ( map_de4839670203443761565_a_b_a
    = ( ^ [K: product_prod_a_b,V: a,F2: a > a,M3: mappin2365780136655028071_a_b_a] : ( map_en8135975850489418638_a_b_a @ K @ F2 @ ( defaul4527334963600521749_a_b_a @ K @ V @ M3 ) ) ) ) ).

% map_default_def
thf(fact_787_lookup__map__default__neq,axiom,
    ! [X: produc7822699449245016212_a_b_a,Y5: produc7822699449245016212_a_b_a,D2: a,F: a > a,M: mappin4996136453398522101_b_a_a] :
      ( ( X != Y5 )
     => ( ( lookup1409243802988840364_b_a_a @ ( map_de3308694818481590113_b_a_a @ X @ D2 @ F @ M ) @ Y5 )
        = ( lookup1409243802988840364_b_a_a @ M @ Y5 ) ) ) ).

% lookup_map_default_neq
thf(fact_788_lookup__map__default__neq,axiom,
    ! [X: list_P6684558232516339260_d_nat,Y5: list_P6684558232516339260_d_nat,D2: mappin2365780136655028071_a_b_a,F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( X != Y5 )
     => ( ( lookup4661578797684577249_a_b_a @ ( map_de582901244306572012_a_b_a @ X @ D2 @ F @ M ) @ Y5 )
        = ( lookup4661578797684577249_a_b_a @ M @ Y5 ) ) ) ).

% lookup_map_default_neq
thf(fact_789_lookup__map__default__neq,axiom,
    ! [X: a,Y5: a,D2: produc5456268773499131775_d_nat,F: produc5456268773499131775_d_nat > produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( X != Y5 )
     => ( ( lookup8480750236328838337_d_nat @ ( map_de521311723871066358_d_nat @ X @ D2 @ F @ M ) @ Y5 )
        = ( lookup8480750236328838337_d_nat @ M @ Y5 ) ) ) ).

% lookup_map_default_neq
thf(fact_790_lookup__map__default__neq,axiom,
    ! [X: a,Y5: a,D2: mappin2365780136655028071_a_b_a,F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( X != Y5 )
     => ( ( lookup1261341372242361769_a_b_a @ ( map_de1440600275304793182_a_b_a @ X @ D2 @ F @ M ) @ Y5 )
        = ( lookup1261341372242361769_a_b_a @ M @ Y5 ) ) ) ).

% lookup_map_default_neq
thf(fact_791_lookup__map__default__neq,axiom,
    ! [X: a,Y5: a,D2: b,F: b > b,M: mapping_a_b] :
      ( ( X != Y5 )
     => ( ( lookup_a_b @ ( map_default_a_b2 @ X @ D2 @ F @ M ) @ Y5 )
        = ( lookup_a_b @ M @ Y5 ) ) ) ).

% lookup_map_default_neq
thf(fact_792_lookup__map__default__neq,axiom,
    ! [X: product_prod_a_b,Y5: product_prod_a_b,D2: a,F: a > a,M: mappin2365780136655028071_a_b_a] :
      ( ( X != Y5 )
     => ( ( lookup863726980359885586_a_b_a @ ( map_de4839670203443761565_a_b_a @ X @ D2 @ F @ M ) @ Y5 )
        = ( lookup863726980359885586_a_b_a @ M @ Y5 ) ) ) ).

% lookup_map_default_neq
thf(fact_793_lookup__default__neq,axiom,
    ! [X: produc7822699449245016212_a_b_a,Y5: produc7822699449245016212_a_b_a,D2: a,M: mappin4996136453398522101_b_a_a] :
      ( ( X != Y5 )
     => ( ( lookup1409243802988840364_b_a_a @ ( defaul2000508409559777513_b_a_a @ X @ D2 @ M ) @ Y5 )
        = ( lookup1409243802988840364_b_a_a @ M @ Y5 ) ) ) ).

% lookup_default_neq
thf(fact_794_lookup__default__neq,axiom,
    ! [X: list_P6684558232516339260_d_nat,Y5: list_P6684558232516339260_d_nat,D2: mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( X != Y5 )
     => ( ( lookup4661578797684577249_a_b_a @ ( defaul2939598897860386660_a_b_a @ X @ D2 @ M ) @ Y5 )
        = ( lookup4661578797684577249_a_b_a @ M @ Y5 ) ) ) ).

% lookup_default_neq
thf(fact_795_lookup__default__neq,axiom,
    ! [X: a,Y5: a,D2: produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( X != Y5 )
     => ( ( lookup8480750236328838337_d_nat @ ( defaul4873115978101149310_d_nat @ X @ D2 @ M ) @ Y5 )
        = ( lookup8480750236328838337_d_nat @ M @ Y5 ) ) ) ).

% lookup_default_neq
thf(fact_796_lookup__default__neq,axiom,
    ! [X: a,Y5: a,D2: mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( X != Y5 )
     => ( ( lookup1261341372242361769_a_b_a @ ( defaul1545336500834713062_a_b_a @ X @ D2 @ M ) @ Y5 )
        = ( lookup1261341372242361769_a_b_a @ M @ Y5 ) ) ) ).

% lookup_default_neq
thf(fact_797_lookup__default__neq,axiom,
    ! [X: a,Y5: a,D2: b,M: mapping_a_b] :
      ( ( X != Y5 )
     => ( ( lookup_a_b @ ( default_a_b @ X @ D2 @ M ) @ Y5 )
        = ( lookup_a_b @ M @ Y5 ) ) ) ).

% lookup_default_neq
thf(fact_798_lookup__default__neq,axiom,
    ! [X: product_prod_a_b,Y5: product_prod_a_b,D2: a,M: mappin2365780136655028071_a_b_a] :
      ( ( X != Y5 )
     => ( ( lookup863726980359885586_a_b_a @ ( defaul4527334963600521749_a_b_a @ X @ D2 @ M ) @ Y5 )
        = ( lookup863726980359885586_a_b_a @ M @ Y5 ) ) ) ).

% lookup_default_neq
thf(fact_799_drop__cur__def,axiom,
    ( drop_cur_c_d
    = ( ^ [I: nat] :
          ( produc2082250591533558949_d_nat
          @ ^ [Q5: c,Tstp: option3893003681250311888_d_nat] :
              ( produc5462365204240479225_d_nat @ Q5
              @ ( case_o6579539980594861021_d_nat @ Tstp
                @ ( produc7043124233248942327_d_nat
                  @ ^ [Ts: d,Tp: nat] : ( if_opt5027858148196338570_d_nat @ ( Tp = I ) @ none_P7808822419737304031_d_nat @ Tstp ) )
                @ Tstp ) ) ) ) ) ).

% drop_cur_def
thf(fact_800_map__default_Osimps_I1_J,axiom,
    ! [K2: a,V2: mappin4996136453398522101_b_a_a,F: mappin4996136453398522101_b_a_a > mappin4996136453398522101_b_a_a] :
      ( ( map_de3993714958335962447_b_a_a @ K2 @ V2 @ F @ nil_Pr5740214961507490724_b_a_a )
      = ( cons_P220927179099053908_b_a_a @ ( produc4259011374121840526_b_a_a @ K2 @ V2 ) @ nil_Pr5740214961507490724_b_a_a ) ) ).

% map_default.simps(1)
thf(fact_801_map__default_Osimps_I1_J,axiom,
    ! [K2: a,V2: list_P6684558232516339260_d_nat,F: list_P6684558232516339260_d_nat > list_P6684558232516339260_d_nat] :
      ( ( map_de7697043261294361186_d_nat @ K2 @ V2 @ F @ nil_Pr1577584438861778787_d_nat )
      = ( cons_P260773292763473843_d_nat @ ( produc919746265332163299_d_nat @ K2 @ V2 ) @ nil_Pr1577584438861778787_d_nat ) ) ).

% map_default.simps(1)
thf(fact_802_map__default_Osimps_I1_J,axiom,
    ! [K2: a,V2: list_P2992919015626629540_a_b_a,F: list_P2992919015626629540_a_b_a > list_P2992919015626629540_a_b_a] :
      ( ( map_de6820110810380196042_a_b_a @ K2 @ V2 @ F @ nil_Pr7283556224815412427_a_b_a )
      = ( cons_P2675685553303450907_a_b_a @ ( produc6994869176599099851_a_b_a @ K2 @ V2 ) @ nil_Pr7283556224815412427_a_b_a ) ) ).

% map_default.simps(1)
thf(fact_803_map__default_Osimps_I1_J,axiom,
    ! [K2: a,V2: list_P1467974523212632366od_a_b,F: list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b] :
      ( ( map_de7622822435605585288od_a_b @ K2 @ V2 @ F @ nil_Pr7243433834293086045od_a_b )
      = ( cons_P8483812562729421581od_a_b @ ( produc6643264353547226055od_a_b @ K2 @ V2 ) @ nil_Pr7243433834293086045od_a_b ) ) ).

% map_default.simps(1)
thf(fact_804_map__default_Osimps_I1_J,axiom,
    ! [K2: a,V2: a,F: a > a] :
      ( ( map_default_a_a @ K2 @ V2 @ F @ nil_Product_prod_a_a )
      = ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ K2 @ V2 ) @ nil_Product_prod_a_a ) ) ).

% map_default.simps(1)
thf(fact_805_map__default_Osimps_I1_J,axiom,
    ! [K2: a,V2: produc5456268773499131775_d_nat,F: produc5456268773499131775_d_nat > produc5456268773499131775_d_nat] :
      ( ( map_de5562044485640834213_d_nat @ K2 @ V2 @ F @ nil_Pr2300798825815202086_d_nat )
      = ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K2 @ V2 ) @ nil_Pr2300798825815202086_d_nat ) ) ).

% map_default.simps(1)
thf(fact_806_map__default_Osimps_I1_J,axiom,
    ! [K2: a,V2: b,F: b > b] :
      ( ( map_default_a_b @ K2 @ V2 @ F @ nil_Product_prod_a_b )
      = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ nil_Product_prod_a_b ) ) ).

% map_default.simps(1)
thf(fact_807_map__default_Osimps_I1_J,axiom,
    ! [K2: produc1990351791824763820_d_nat,V2: mappin2365780136655028071_a_b_a,F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a] :
      ( ( map_de3625967655130690285_a_b_a @ K2 @ V2 @ F @ nil_Pr3763312433997567610_a_b_a )
      = ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K2 @ V2 ) @ nil_Pr3763312433997567610_a_b_a ) ) ).

% map_default.simps(1)
thf(fact_808_map__default_Osimps_I1_J,axiom,
    ! [K2: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a,F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a] :
      ( ( map_de1795163708464626941_a_b_a @ K2 @ V2 @ F @ nil_Pr1908111780246848906_a_b_a )
      = ( cons_P2804281896780507450_a_b_a @ ( produc2616710729536263484_a_b_a @ K2 @ V2 ) @ nil_Pr1908111780246848906_a_b_a ) ) ).

% map_default.simps(1)
thf(fact_809_map__default_Osimps_I1_J,axiom,
    ! [K2: a,V2: mappin2365780136655028071_a_b_a,F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a] :
      ( ( map_de7165279943232214925_a_b_a @ K2 @ V2 @ F @ nil_Pr7632893056509733646_a_b_a )
      = ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K2 @ V2 ) @ nil_Pr7632893056509733646_a_b_a ) ) ).

% map_default.simps(1)
thf(fact_810_delete__aux_Oelims,axiom,
    ! [X: a,Xa: list_P3594502398639369124_b_a_a,Y5: list_P3594502398639369124_b_a_a] :
      ( ( ( delete3482169990200181533_b_a_a @ X @ Xa )
        = Y5 )
     => ( ( ( Xa = nil_Pr5740214961507490724_b_a_a )
         => ( Y5 != nil_Pr5740214961507490724_b_a_a ) )
       => ~ ! [K6: a,V4: mappin4996136453398522101_b_a_a,Xs: list_P3594502398639369124_b_a_a] :
              ( ( Xa
                = ( cons_P220927179099053908_b_a_a @ ( produc4259011374121840526_b_a_a @ K6 @ V4 ) @ Xs ) )
             => ~ ( ( ( X = K6 )
                   => ( Y5 = Xs ) )
                  & ( ( X != K6 )
                   => ( Y5
                      = ( cons_P220927179099053908_b_a_a @ ( produc4259011374121840526_b_a_a @ K6 @ V4 ) @ ( delete3482169990200181533_b_a_a @ X @ Xs ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_811_delete__aux_Oelims,axiom,
    ! [X: a,Xa: list_P9063253934932097401_d_nat,Y5: list_P9063253934932097401_d_nat] :
      ( ( ( delete6171817490833448468_d_nat @ X @ Xa )
        = Y5 )
     => ( ( ( Xa = nil_Pr1577584438861778787_d_nat )
         => ( Y5 != nil_Pr1577584438861778787_d_nat ) )
       => ~ ! [K6: a,V4: list_P6684558232516339260_d_nat,Xs: list_P9063253934932097401_d_nat] :
              ( ( Xa
                = ( cons_P260773292763473843_d_nat @ ( produc919746265332163299_d_nat @ K6 @ V4 ) @ Xs ) )
             => ~ ( ( ( X = K6 )
                   => ( Y5 = Xs ) )
                  & ( ( X != K6 )
                   => ( Y5
                      = ( cons_P260773292763473843_d_nat @ ( produc919746265332163299_d_nat @ K6 @ V4 ) @ ( delete6171817490833448468_d_nat @ X @ Xs ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_812_delete__aux_Oelims,axiom,
    ! [X: a,Xa: list_P3172577687508813921_a_b_a,Y5: list_P3172577687508813921_a_b_a] :
      ( ( ( delete4222590184415954300_a_b_a @ X @ Xa )
        = Y5 )
     => ( ( ( Xa = nil_Pr7283556224815412427_a_b_a )
         => ( Y5 != nil_Pr7283556224815412427_a_b_a ) )
       => ~ ! [K6: a,V4: list_P2992919015626629540_a_b_a,Xs: list_P3172577687508813921_a_b_a] :
              ( ( Xa
                = ( cons_P2675685553303450907_a_b_a @ ( produc6994869176599099851_a_b_a @ K6 @ V4 ) @ Xs ) )
             => ~ ( ( ( X = K6 )
                   => ( Y5 = Xs ) )
                  & ( ( X != K6 )
                   => ( Y5
                      = ( cons_P2675685553303450907_a_b_a @ ( produc6994869176599099851_a_b_a @ K6 @ V4 ) @ ( delete4222590184415954300_a_b_a @ X @ Xs ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_813_delete__aux_Oelims,axiom,
    ! [X: a,Xa: list_P5700792096213675741od_a_b,Y5: list_P5700792096213675741od_a_b] :
      ( ( ( delete3143381019966442326od_a_b @ X @ Xa )
        = Y5 )
     => ( ( ( Xa = nil_Pr7243433834293086045od_a_b )
         => ( Y5 != nil_Pr7243433834293086045od_a_b ) )
       => ~ ! [K6: a,V4: list_P1467974523212632366od_a_b,Xs: list_P5700792096213675741od_a_b] :
              ( ( Xa
                = ( cons_P8483812562729421581od_a_b @ ( produc6643264353547226055od_a_b @ K6 @ V4 ) @ Xs ) )
             => ~ ( ( ( X = K6 )
                   => ( Y5 = Xs ) )
                  & ( ( X != K6 )
                   => ( Y5
                      = ( cons_P8483812562729421581od_a_b @ ( produc6643264353547226055od_a_b @ K6 @ V4 ) @ ( delete3143381019966442326od_a_b @ X @ Xs ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_814_delete__aux_Oelims,axiom,
    ! [X: a,Xa: list_P1396940483166286381od_a_a,Y5: list_P1396940483166286381od_a_a] :
      ( ( ( delete_aux_a_a @ X @ Xa )
        = Y5 )
     => ( ( ( Xa = nil_Product_prod_a_a )
         => ( Y5 != nil_Product_prod_a_a ) )
       => ~ ! [K6: a,V4: a,Xs: list_P1396940483166286381od_a_a] :
              ( ( Xa
                = ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ K6 @ V4 ) @ Xs ) )
             => ~ ( ( ( X = K6 )
                   => ( Y5 = Xs ) )
                  & ( ( X != K6 )
                   => ( Y5
                      = ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ K6 @ V4 ) @ ( delete_aux_a_a @ X @ Xs ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_815_delete__aux_Oelims,axiom,
    ! [X: a,Xa: list_P6684558232516339260_d_nat,Y5: list_P6684558232516339260_d_nat] :
      ( ( ( delete6174134448749622103_d_nat @ X @ Xa )
        = Y5 )
     => ( ( ( Xa = nil_Pr2300798825815202086_d_nat )
         => ( Y5 != nil_Pr2300798825815202086_d_nat ) )
       => ~ ! [K6: a,V4: produc5456268773499131775_d_nat,Xs: list_P6684558232516339260_d_nat] :
              ( ( Xa
                = ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K6 @ V4 ) @ Xs ) )
             => ~ ( ( ( X = K6 )
                   => ( Y5 = Xs ) )
                  & ( ( X != K6 )
                   => ( Y5
                      = ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K6 @ V4 ) @ ( delete6174134448749622103_d_nat @ X @ Xs ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_816_delete__aux_Oelims,axiom,
    ! [X: a,Xa: list_P1467974523212632366od_a_b,Y5: list_P1467974523212632366od_a_b] :
      ( ( ( delete_aux_a_b @ X @ Xa )
        = Y5 )
     => ( ( ( Xa = nil_Product_prod_a_b )
         => ( Y5 != nil_Product_prod_a_b ) )
       => ~ ! [K6: a,V4: b,Xs: list_P1467974523212632366od_a_b] :
              ( ( Xa
                = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K6 @ V4 ) @ Xs ) )
             => ~ ( ( ( X = K6 )
                   => ( Y5 = Xs ) )
                  & ( ( X != K6 )
                   => ( Y5
                      = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K6 @ V4 ) @ ( delete_aux_a_b @ X @ Xs ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_817_delete__aux_Oelims,axiom,
    ! [X: produc1990351791824763820_d_nat,Xa: list_P4793483941206178810_a_b_a,Y5: list_P4793483941206178810_a_b_a] :
      ( ( ( delete7618755331649393595_a_b_a @ X @ Xa )
        = Y5 )
     => ( ( ( Xa = nil_Pr3763312433997567610_a_b_a )
         => ( Y5 != nil_Pr3763312433997567610_a_b_a ) )
       => ~ ! [K6: produc1990351791824763820_d_nat,V4: mappin2365780136655028071_a_b_a,Xs: list_P4793483941206178810_a_b_a] :
              ( ( Xa
                = ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K6 @ V4 ) @ Xs ) )
             => ~ ( ( ( X = K6 )
                   => ( Y5 = Xs ) )
                  & ( ( X != K6 )
                   => ( Y5
                      = ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K6 @ V4 ) @ ( delete7618755331649393595_a_b_a @ X @ Xs ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_818_delete__aux_Oelims,axiom,
    ! [X: list_P6684558232516339260_d_nat,Xa: list_P7611574810425128074_a_b_a,Y5: list_P7611574810425128074_a_b_a] :
      ( ( ( delete3219952628631576267_a_b_a @ X @ Xa )
        = Y5 )
     => ( ( ( Xa = nil_Pr1908111780246848906_a_b_a )
         => ( Y5 != nil_Pr1908111780246848906_a_b_a ) )
       => ~ ! [K6: list_P6684558232516339260_d_nat,V4: mappin2365780136655028071_a_b_a,Xs: list_P7611574810425128074_a_b_a] :
              ( ( Xa
                = ( cons_P2804281896780507450_a_b_a @ ( produc2616710729536263484_a_b_a @ K6 @ V4 ) @ Xs ) )
             => ~ ( ( ( X = K6 )
                   => ( Y5 = Xs ) )
                  & ( ( X != K6 )
                   => ( Y5
                      = ( cons_P2804281896780507450_a_b_a @ ( produc2616710729536263484_a_b_a @ K6 @ V4 ) @ ( delete3219952628631576267_a_b_a @ X @ Xs ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_819_delete__aux_Oelims,axiom,
    ! [X: a,Xa: list_P2992919015626629540_a_b_a,Y5: list_P2992919015626629540_a_b_a] :
      ( ( ( delete1755771892824606527_a_b_a @ X @ Xa )
        = Y5 )
     => ( ( ( Xa = nil_Pr7632893056509733646_a_b_a )
         => ( Y5 != nil_Pr7632893056509733646_a_b_a ) )
       => ~ ! [K6: a,V4: mappin2365780136655028071_a_b_a,Xs: list_P2992919015626629540_a_b_a] :
              ( ( Xa
                = ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K6 @ V4 ) @ Xs ) )
             => ~ ( ( ( X = K6 )
                   => ( Y5 = Xs ) )
                  & ( ( X != K6 )
                   => ( Y5
                      = ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K6 @ V4 ) @ ( delete1755771892824606527_a_b_a @ X @ Xs ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_820_delete__aux__eq__Nil__conv,axiom,
    ! [K2: a,Ts2: list_P3594502398639369124_b_a_a] :
      ( ( ( delete3482169990200181533_b_a_a @ K2 @ Ts2 )
        = nil_Pr5740214961507490724_b_a_a )
      = ( ( Ts2 = nil_Pr5740214961507490724_b_a_a )
        | ? [V: mappin4996136453398522101_b_a_a] :
            ( Ts2
            = ( cons_P220927179099053908_b_a_a @ ( produc4259011374121840526_b_a_a @ K2 @ V ) @ nil_Pr5740214961507490724_b_a_a ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_821_delete__aux__eq__Nil__conv,axiom,
    ! [K2: a,Ts2: list_P9063253934932097401_d_nat] :
      ( ( ( delete6171817490833448468_d_nat @ K2 @ Ts2 )
        = nil_Pr1577584438861778787_d_nat )
      = ( ( Ts2 = nil_Pr1577584438861778787_d_nat )
        | ? [V: list_P6684558232516339260_d_nat] :
            ( Ts2
            = ( cons_P260773292763473843_d_nat @ ( produc919746265332163299_d_nat @ K2 @ V ) @ nil_Pr1577584438861778787_d_nat ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_822_delete__aux__eq__Nil__conv,axiom,
    ! [K2: a,Ts2: list_P3172577687508813921_a_b_a] :
      ( ( ( delete4222590184415954300_a_b_a @ K2 @ Ts2 )
        = nil_Pr7283556224815412427_a_b_a )
      = ( ( Ts2 = nil_Pr7283556224815412427_a_b_a )
        | ? [V: list_P2992919015626629540_a_b_a] :
            ( Ts2
            = ( cons_P2675685553303450907_a_b_a @ ( produc6994869176599099851_a_b_a @ K2 @ V ) @ nil_Pr7283556224815412427_a_b_a ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_823_delete__aux__eq__Nil__conv,axiom,
    ! [K2: a,Ts2: list_P5700792096213675741od_a_b] :
      ( ( ( delete3143381019966442326od_a_b @ K2 @ Ts2 )
        = nil_Pr7243433834293086045od_a_b )
      = ( ( Ts2 = nil_Pr7243433834293086045od_a_b )
        | ? [V: list_P1467974523212632366od_a_b] :
            ( Ts2
            = ( cons_P8483812562729421581od_a_b @ ( produc6643264353547226055od_a_b @ K2 @ V ) @ nil_Pr7243433834293086045od_a_b ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_824_delete__aux__eq__Nil__conv,axiom,
    ! [K2: a,Ts2: list_P1396940483166286381od_a_a] :
      ( ( ( delete_aux_a_a @ K2 @ Ts2 )
        = nil_Product_prod_a_a )
      = ( ( Ts2 = nil_Product_prod_a_a )
        | ? [V: a] :
            ( Ts2
            = ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ K2 @ V ) @ nil_Product_prod_a_a ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_825_delete__aux__eq__Nil__conv,axiom,
    ! [K2: a,Ts2: list_P6684558232516339260_d_nat] :
      ( ( ( delete6174134448749622103_d_nat @ K2 @ Ts2 )
        = nil_Pr2300798825815202086_d_nat )
      = ( ( Ts2 = nil_Pr2300798825815202086_d_nat )
        | ? [V: produc5456268773499131775_d_nat] :
            ( Ts2
            = ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K2 @ V ) @ nil_Pr2300798825815202086_d_nat ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_826_delete__aux__eq__Nil__conv,axiom,
    ! [K2: a,Ts2: list_P1467974523212632366od_a_b] :
      ( ( ( delete_aux_a_b @ K2 @ Ts2 )
        = nil_Product_prod_a_b )
      = ( ( Ts2 = nil_Product_prod_a_b )
        | ? [V: b] :
            ( Ts2
            = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V ) @ nil_Product_prod_a_b ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_827_delete__aux__eq__Nil__conv,axiom,
    ! [K2: produc1990351791824763820_d_nat,Ts2: list_P4793483941206178810_a_b_a] :
      ( ( ( delete7618755331649393595_a_b_a @ K2 @ Ts2 )
        = nil_Pr3763312433997567610_a_b_a )
      = ( ( Ts2 = nil_Pr3763312433997567610_a_b_a )
        | ? [V: mappin2365780136655028071_a_b_a] :
            ( Ts2
            = ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K2 @ V ) @ nil_Pr3763312433997567610_a_b_a ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_828_delete__aux__eq__Nil__conv,axiom,
    ! [K2: list_P6684558232516339260_d_nat,Ts2: list_P7611574810425128074_a_b_a] :
      ( ( ( delete3219952628631576267_a_b_a @ K2 @ Ts2 )
        = nil_Pr1908111780246848906_a_b_a )
      = ( ( Ts2 = nil_Pr1908111780246848906_a_b_a )
        | ? [V: mappin2365780136655028071_a_b_a] :
            ( Ts2
            = ( cons_P2804281896780507450_a_b_a @ ( produc2616710729536263484_a_b_a @ K2 @ V ) @ nil_Pr1908111780246848906_a_b_a ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_829_delete__aux__eq__Nil__conv,axiom,
    ! [K2: a,Ts2: list_P2992919015626629540_a_b_a] :
      ( ( ( delete1755771892824606527_a_b_a @ K2 @ Ts2 )
        = nil_Pr7632893056509733646_a_b_a )
      = ( ( Ts2 = nil_Pr7632893056509733646_a_b_a )
        | ? [V: mappin2365780136655028071_a_b_a] :
            ( Ts2
            = ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K2 @ V ) @ nil_Pr7632893056509733646_a_b_a ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_830_listrel_Ocases,axiom,
    ! [A1: list_a,A22: list_b,R3: set_Product_prod_a_b] :
      ( ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ A1 @ A22 ) @ ( listrel_a_b @ R3 ) )
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_b ) )
       => ~ ! [X3: a,Y3: b,Xs: list_a] :
              ( ( A1
                = ( cons_a @ X3 @ Xs ) )
             => ! [Ys: list_b] :
                  ( ( A22
                    = ( cons_b @ Y3 @ Ys ) )
                 => ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X3 @ Y3 ) @ R3 )
                   => ~ ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ Xs @ Ys ) @ ( listrel_a_b @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_831_listrel_Ocases,axiom,
    ! [A1: list_a,A22: list_m9210641325446214647_a_b_a,R3: set_Pr2084289749507318346_a_b_a] :
      ( ( member5215947914154143169_a_b_a @ ( produc1462633373383792292_a_b_a @ A1 @ A22 ) @ ( listre3604961095617705268_a_b_a @ R3 ) )
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_ma546712388551201761_a_b_a ) )
       => ~ ! [X3: a,Y3: mappin2365780136655028071_a_b_a,Xs: list_a] :
              ( ( A1
                = ( cons_a @ X3 @ Xs ) )
             => ! [Ys: list_m9210641325446214647_a_b_a] :
                  ( ( A22
                    = ( cons_m8574943908289518641_a_b_a @ Y3 @ Ys ) )
                 => ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) @ R3 )
                   => ~ ( member5215947914154143169_a_b_a @ ( produc1462633373383792292_a_b_a @ Xs @ Ys ) @ ( listre3604961095617705268_a_b_a @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_832_listrel_Ocases,axiom,
    ! [A1: list_a,A22: list_P7053655179267667343_d_nat,R3: set_Pr8583615357880026082_d_nat] :
      ( ( member7164185383468370905_d_nat @ ( produc6231069337089993148_d_nat @ A1 @ A22 ) @ ( listre7853655052829806284_d_nat @ R3 ) )
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_Pr3001256489040166649_d_nat ) )
       => ~ ! [X3: a,Y3: produc5456268773499131775_d_nat,Xs: list_a] :
              ( ( A1
                = ( cons_a @ X3 @ Xs ) )
             => ! [Ys: list_P7053655179267667343_d_nat] :
                  ( ( A22
                    = ( cons_P8522577077107056457_d_nat @ Y3 @ Ys ) )
                 => ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X3 @ Y3 ) @ R3 )
                   => ~ ( member7164185383468370905_d_nat @ ( produc6231069337089993148_d_nat @ Xs @ Ys ) @ ( listre7853655052829806284_d_nat @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_833_listrel_Ocases,axiom,
    ! [A1: list_P6684558232516339260_d_nat,A22: list_m9210641325446214647_a_b_a,R3: set_Pr8226792695654363348_a_b_a] :
      ( ( member2359517100237508157_a_b_a @ ( produc6307639320882748236_a_b_a @ A1 @ A22 ) @ ( listre3191202743213796614_a_b_a @ R3 ) )
     => ( ( ( A1 = nil_Pr2300798825815202086_d_nat )
         => ( A22 != nil_ma546712388551201761_a_b_a ) )
       => ~ ! [X3: produc1990351791824763820_d_nat,Y3: mappin2365780136655028071_a_b_a,Xs: list_P6684558232516339260_d_nat] :
              ( ( A1
                = ( cons_P2594832900118292854_d_nat @ X3 @ Xs ) )
             => ! [Ys: list_m9210641325446214647_a_b_a] :
                  ( ( A22
                    = ( cons_m8574943908289518641_a_b_a @ Y3 @ Ys ) )
                 => ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X3 @ Y3 ) @ R3 )
                   => ~ ( member2359517100237508157_a_b_a @ ( produc6307639320882748236_a_b_a @ Xs @ Ys ) @ ( listre3191202743213796614_a_b_a @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_834_listrel_Ocases,axiom,
    ! [A1: list_l337834041258660428_d_nat,A22: list_m9210641325446214647_a_b_a,R3: set_Pr2043227878544749668_a_b_a] :
      ( ( member3296818671409482061_a_b_a @ ( produc3873751758850487132_a_b_a @ A1 @ A22 ) @ ( listre6449073666664715926_a_b_a @ R3 ) )
     => ( ( ( A1 = nil_li1008261075667615414_d_nat )
         => ( A22 != nil_ma546712388551201761_a_b_a ) )
       => ~ ! [X3: list_P6684558232516339260_d_nat,Y3: mappin2365780136655028071_a_b_a,Xs: list_l337834041258660428_d_nat] :
              ( ( A1
                = ( cons_l4410424432949683974_d_nat @ X3 @ Xs ) )
             => ! [Ys: list_m9210641325446214647_a_b_a] :
                  ( ( A22
                    = ( cons_m8574943908289518641_a_b_a @ Y3 @ Ys ) )
                 => ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X3 @ Y3 ) @ R3 )
                   => ~ ( member3296818671409482061_a_b_a @ ( produc3873751758850487132_a_b_a @ Xs @ Ys ) @ ( listre6449073666664715926_a_b_a @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_835_listrel_Ocases,axiom,
    ! [A1: list_b,A22: list_b,R3: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ A1 @ A22 ) @ ( listrel_b_b @ R3 ) )
     => ( ( ( A1 = nil_b )
         => ( A22 != nil_b ) )
       => ~ ! [X3: b,Y3: b,Xs: list_b] :
              ( ( A1
                = ( cons_b @ X3 @ Xs ) )
             => ! [Ys: list_b] :
                  ( ( A22
                    = ( cons_b @ Y3 @ Ys ) )
                 => ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R3 )
                   => ~ ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs @ Ys ) @ ( listrel_b_b @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_836_listrel_Ocases,axiom,
    ! [A1: list_b,A22: list_a,R3: set_Product_prod_b_a] :
      ( ( member2919287837941893007list_a @ ( produc1564554174005236310list_a @ A1 @ A22 ) @ ( listrel_b_a @ R3 ) )
     => ( ( ( A1 = nil_b )
         => ( A22 != nil_a ) )
       => ~ ! [X3: b,Y3: a,Xs: list_b] :
              ( ( A1
                = ( cons_b @ X3 @ Xs ) )
             => ! [Ys: list_a] :
                  ( ( A22
                    = ( cons_a @ Y3 @ Ys ) )
                 => ( ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X3 @ Y3 ) @ R3 )
                   => ~ ( member2919287837941893007list_a @ ( produc1564554174005236310list_a @ Xs @ Ys ) @ ( listrel_b_a @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_837_listrel_Ocases,axiom,
    ! [A1: list_a,A22: list_a,R3: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R3 ) )
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_a ) )
       => ~ ! [X3: a,Y3: a,Xs: list_a] :
              ( ( A1
                = ( cons_a @ X3 @ Xs ) )
             => ! [Ys: list_a] :
                  ( ( A22
                    = ( cons_a @ Y3 @ Ys ) )
                 => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R3 )
                   => ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_838_listrel_Ocases,axiom,
    ! [A1: list_P1467974523212632366od_a_b,A22: list_b,R3: set_Pr2374853161201823710_a_b_b] :
      ( ( member5653424251854706471list_b @ ( produc8422988059101989998list_b @ A1 @ A22 ) @ ( listre4992324826761406024_a_b_b @ R3 ) )
     => ( ( ( A1 = nil_Product_prod_a_b )
         => ( A22 != nil_b ) )
       => ~ ! [X3: product_prod_a_b,Y3: b,Xs: list_P1467974523212632366od_a_b] :
              ( ( A1
                = ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) )
             => ! [Ys: list_b] :
                  ( ( A22
                    = ( cons_b @ Y3 @ Ys ) )
                 => ( ( member2194267521500829095_a_b_b @ ( produc5056887847982525422_a_b_b @ X3 @ Y3 ) @ R3 )
                   => ~ ( member5653424251854706471list_b @ ( produc8422988059101989998list_b @ Xs @ Ys ) @ ( listre4992324826761406024_a_b_b @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_839_listrel_Ocases,axiom,
    ! [A1: list_P1467974523212632366od_a_b,A22: list_a,R3: set_Pr2303819121155477725_a_b_a] :
      ( ( member5582390211808360486list_a @ ( produc8422988054798761197list_a @ A1 @ A22 ) @ ( listre4992324826761406023_a_b_a @ R3 ) )
     => ( ( ( A1 = nil_Product_prod_a_b )
         => ( A22 != nil_a ) )
       => ~ ! [X3: product_prod_a_b,Y3: a,Xs: list_P1467974523212632366od_a_b] :
              ( ( A1
                = ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) )
             => ! [Ys: list_a] :
                  ( ( A22
                    = ( cons_a @ Y3 @ Ys ) )
                 => ( ( member2194267517197600294_a_b_a @ ( produc5056887847982525421_a_b_a @ X3 @ Y3 ) @ R3 )
                   => ~ ( member5582390211808360486list_a @ ( produc8422988054798761197list_a @ Xs @ Ys ) @ ( listre4992324826761406023_a_b_a @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_840_listrel_Osimps,axiom,
    ! [A1: list_a,A22: list_b,R3: set_Product_prod_a_b] :
      ( ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ A1 @ A22 ) @ ( listrel_a_b @ R3 ) )
      = ( ( ( A1 = nil_a )
          & ( A22 = nil_b ) )
        | ? [X4: a,Y4: b,Xs3: list_a,Ys2: list_b] :
            ( ( A1
              = ( cons_a @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_b @ Y4 @ Ys2 ) )
            & ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ R3 )
            & ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ Xs3 @ Ys2 ) @ ( listrel_a_b @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_841_listrel_Osimps,axiom,
    ! [A1: list_a,A22: list_m9210641325446214647_a_b_a,R3: set_Pr2084289749507318346_a_b_a] :
      ( ( member5215947914154143169_a_b_a @ ( produc1462633373383792292_a_b_a @ A1 @ A22 ) @ ( listre3604961095617705268_a_b_a @ R3 ) )
      = ( ( ( A1 = nil_a )
          & ( A22 = nil_ma546712388551201761_a_b_a ) )
        | ? [X4: a,Y4: mappin2365780136655028071_a_b_a,Xs3: list_a,Ys2: list_m9210641325446214647_a_b_a] :
            ( ( A1
              = ( cons_a @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_m8574943908289518641_a_b_a @ Y4 @ Ys2 ) )
            & ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ R3 )
            & ( member5215947914154143169_a_b_a @ ( produc1462633373383792292_a_b_a @ Xs3 @ Ys2 ) @ ( listre3604961095617705268_a_b_a @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_842_listrel_Osimps,axiom,
    ! [A1: list_a,A22: list_P7053655179267667343_d_nat,R3: set_Pr8583615357880026082_d_nat] :
      ( ( member7164185383468370905_d_nat @ ( produc6231069337089993148_d_nat @ A1 @ A22 ) @ ( listre7853655052829806284_d_nat @ R3 ) )
      = ( ( ( A1 = nil_a )
          & ( A22 = nil_Pr3001256489040166649_d_nat ) )
        | ? [X4: a,Y4: produc5456268773499131775_d_nat,Xs3: list_a,Ys2: list_P7053655179267667343_d_nat] :
            ( ( A1
              = ( cons_a @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_P8522577077107056457_d_nat @ Y4 @ Ys2 ) )
            & ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ R3 )
            & ( member7164185383468370905_d_nat @ ( produc6231069337089993148_d_nat @ Xs3 @ Ys2 ) @ ( listre7853655052829806284_d_nat @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_843_listrel_Osimps,axiom,
    ! [A1: list_P6684558232516339260_d_nat,A22: list_m9210641325446214647_a_b_a,R3: set_Pr8226792695654363348_a_b_a] :
      ( ( member2359517100237508157_a_b_a @ ( produc6307639320882748236_a_b_a @ A1 @ A22 ) @ ( listre3191202743213796614_a_b_a @ R3 ) )
      = ( ( ( A1 = nil_Pr2300798825815202086_d_nat )
          & ( A22 = nil_ma546712388551201761_a_b_a ) )
        | ? [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a,Xs3: list_P6684558232516339260_d_nat,Ys2: list_m9210641325446214647_a_b_a] :
            ( ( A1
              = ( cons_P2594832900118292854_d_nat @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_m8574943908289518641_a_b_a @ Y4 @ Ys2 ) )
            & ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ R3 )
            & ( member2359517100237508157_a_b_a @ ( produc6307639320882748236_a_b_a @ Xs3 @ Ys2 ) @ ( listre3191202743213796614_a_b_a @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_844_listrel_Osimps,axiom,
    ! [A1: list_l337834041258660428_d_nat,A22: list_m9210641325446214647_a_b_a,R3: set_Pr2043227878544749668_a_b_a] :
      ( ( member3296818671409482061_a_b_a @ ( produc3873751758850487132_a_b_a @ A1 @ A22 ) @ ( listre6449073666664715926_a_b_a @ R3 ) )
      = ( ( ( A1 = nil_li1008261075667615414_d_nat )
          & ( A22 = nil_ma546712388551201761_a_b_a ) )
        | ? [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a,Xs3: list_l337834041258660428_d_nat,Ys2: list_m9210641325446214647_a_b_a] :
            ( ( A1
              = ( cons_l4410424432949683974_d_nat @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_m8574943908289518641_a_b_a @ Y4 @ Ys2 ) )
            & ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ R3 )
            & ( member3296818671409482061_a_b_a @ ( produc3873751758850487132_a_b_a @ Xs3 @ Ys2 ) @ ( listre6449073666664715926_a_b_a @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_845_listrel_Osimps,axiom,
    ! [A1: list_b,A22: list_b,R3: set_Product_prod_b_b] :
      ( ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ A1 @ A22 ) @ ( listrel_b_b @ R3 ) )
      = ( ( ( A1 = nil_b )
          & ( A22 = nil_b ) )
        | ? [X4: b,Y4: b,Xs3: list_b,Ys2: list_b] :
            ( ( A1
              = ( cons_b @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_b @ Y4 @ Ys2 ) )
            & ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R3 )
            & ( member2990321877988238992list_b @ ( produc1564554178308465111list_b @ Xs3 @ Ys2 ) @ ( listrel_b_b @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_846_listrel_Osimps,axiom,
    ! [A1: list_b,A22: list_a,R3: set_Product_prod_b_a] :
      ( ( member2919287837941893007list_a @ ( produc1564554174005236310list_a @ A1 @ A22 ) @ ( listrel_b_a @ R3 ) )
      = ( ( ( A1 = nil_b )
          & ( A22 = nil_a ) )
        | ? [X4: b,Y4: a,Xs3: list_b,Ys2: list_a] :
            ( ( A1
              = ( cons_b @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_a @ Y4 @ Ys2 ) )
            & ( member7862447932407534991od_b_a @ ( product_Pair_b_a @ X4 @ Y4 ) @ R3 )
            & ( member2919287837941893007list_a @ ( produc1564554174005236310list_a @ Xs3 @ Ys2 ) @ ( listrel_b_a @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_847_listrel_Osimps,axiom,
    ! [A1: list_a,A22: list_a,R3: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R3 ) )
      = ( ( ( A1 = nil_a )
          & ( A22 = nil_a ) )
        | ? [X4: a,Y4: a,Xs3: list_a,Ys2: list_a] :
            ( ( A1
              = ( cons_a @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_a @ Y4 @ Ys2 ) )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R3 )
            & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys2 ) @ ( listrel_a_a @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_848_listrel_Osimps,axiom,
    ! [A1: list_P1467974523212632366od_a_b,A22: list_b,R3: set_Pr2374853161201823710_a_b_b] :
      ( ( member5653424251854706471list_b @ ( produc8422988059101989998list_b @ A1 @ A22 ) @ ( listre4992324826761406024_a_b_b @ R3 ) )
      = ( ( ( A1 = nil_Product_prod_a_b )
          & ( A22 = nil_b ) )
        | ? [X4: product_prod_a_b,Y4: b,Xs3: list_P1467974523212632366od_a_b,Ys2: list_b] :
            ( ( A1
              = ( cons_P7316939131009794654od_a_b @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_b @ Y4 @ Ys2 ) )
            & ( member2194267521500829095_a_b_b @ ( produc5056887847982525422_a_b_b @ X4 @ Y4 ) @ R3 )
            & ( member5653424251854706471list_b @ ( produc8422988059101989998list_b @ Xs3 @ Ys2 ) @ ( listre4992324826761406024_a_b_b @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_849_listrel_Osimps,axiom,
    ! [A1: list_P1467974523212632366od_a_b,A22: list_a,R3: set_Pr2303819121155477725_a_b_a] :
      ( ( member5582390211808360486list_a @ ( produc8422988054798761197list_a @ A1 @ A22 ) @ ( listre4992324826761406023_a_b_a @ R3 ) )
      = ( ( ( A1 = nil_Product_prod_a_b )
          & ( A22 = nil_a ) )
        | ? [X4: product_prod_a_b,Y4: a,Xs3: list_P1467974523212632366od_a_b,Ys2: list_a] :
            ( ( A1
              = ( cons_P7316939131009794654od_a_b @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_a @ Y4 @ Ys2 ) )
            & ( member2194267517197600294_a_b_a @ ( produc5056887847982525421_a_b_a @ X4 @ Y4 ) @ R3 )
            & ( member5582390211808360486list_a @ ( produc8422988054798761197list_a @ Xs3 @ Ys2 ) @ ( listre4992324826761406023_a_b_a @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_850_in__entriesI,axiom,
    ! [M: mapping_a_b,K2: a,V2: b] :
      ( ( ( lookup_a_b @ M @ K2 )
        = ( some_b @ V2 ) )
     => ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ ( entries_a_b @ M ) ) ) ).

% in_entriesI
thf(fact_851_in__entriesI,axiom,
    ! [M: mappin2365780136655028071_a_b_a,K2: product_prod_a_b,V2: a] :
      ( ( ( lookup863726980359885586_a_b_a @ M @ K2 )
        = ( some_a @ V2 ) )
     => ( member2194267517197600294_a_b_a @ ( produc5056887847982525421_a_b_a @ K2 @ V2 ) @ ( entrie935192368317430852_a_b_a @ M ) ) ) ).

% in_entriesI
thf(fact_852_in__entriesI,axiom,
    ! [M: mappin4519906018992269226_a_b_a,K2: a,V2: mappin2365780136655028071_a_b_a] :
      ( ( ( lookup1261341372242361769_a_b_a @ M @ K2 )
        = ( some_m5820982296856387138_a_b_a @ V2 ) )
     => ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ K2 @ V2 ) @ ( entrie7794251174888650743_a_b_a @ M ) ) ) ).

% in_entriesI
thf(fact_853_in__entriesI,axiom,
    ! [M: mappin6535725686296717250_d_nat,K2: a,V2: produc5456268773499131775_d_nat] :
      ( ( ( lookup8480750236328838337_d_nat @ M @ K2 )
        = ( some_P3314015041498521818_d_nat @ V2 ) )
     => ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ K2 @ V2 ) @ ( entrie8866367667353307151_d_nat @ M ) ) ) ).

% in_entriesI
thf(fact_854_in__entriesI,axiom,
    ! [M: mappin1208405706615879902_a_b_a,K2: produc1990351791824763820_d_nat,V2: mappin2365780136655028071_a_b_a] :
      ( ( ( lookup2444594617825954769_a_b_a @ M @ K2 )
        = ( some_m5820982296856387138_a_b_a @ V2 ) )
     => ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ K2 @ V2 ) @ ( entrie369595780118757635_a_b_a @ M ) ) ) ).

% in_entriesI
thf(fact_855_in__entriesI,axiom,
    ! [M: mappin187059948859696622_a_b_a,K2: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a] :
      ( ( ( lookup4661578797684577249_a_b_a @ M @ K2 )
        = ( some_m5820982296856387138_a_b_a @ V2 ) )
     => ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ K2 @ V2 ) @ ( entrie4359402741916977171_a_b_a @ M ) ) ) ).

% in_entriesI
thf(fact_856_in__entriesI,axiom,
    ! [M: mapping_a_a,K2: a,V2: a] :
      ( ( ( lookup_a_a @ M @ K2 )
        = ( some_a @ V2 ) )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ K2 @ V2 ) @ ( entries_a_a @ M ) ) ) ).

% in_entriesI
thf(fact_857_in__entriesI,axiom,
    ! [M: mappin6513503936400150465od_a_b,K2: a,V2: list_P1467974523212632366od_a_b] :
      ( ( ( lookup611153099767340140od_a_b @ M @ K2 )
        = ( some_l1253897223994727059od_a_b @ V2 ) )
     => ( member8745141763671436032od_a_b @ ( produc6643264353547226055od_a_b @ K2 @ V2 ) @ ( entrie6405578741455881374od_a_b @ M ) ) ) ).

% in_entriesI
thf(fact_858_in__entriesI,axiom,
    ! [M: mappin4996136453398522101_b_a_a,K2: produc7822699449245016212_a_b_a,V2: a] :
      ( ( ( lookup1409243802988840364_b_a_a @ M @ K2 )
        = ( some_a @ V2 ) )
     => ( member8438774017255854454_b_a_a @ ( produc1564263632067424529_b_a_a @ K2 @ V2 ) @ ( entrie4500269644305842682_b_a_a @ M ) ) ) ).

% in_entriesI
thf(fact_859_in__entriesI,axiom,
    ! [M: mappin7679310425283589991_a_b_a,K2: a,V2: list_P2992919015626629540_a_b_a] :
      ( ( ( lookup6501822868788954342_a_b_a @ M @ K2 )
        = ( some_l8092144040308835327_a_b_a @ V2 ) )
     => ( member378967531903691624_a_b_a @ ( produc6994869176599099851_a_b_a @ K2 @ V2 ) @ ( entrie577405926735905332_a_b_a @ M ) ) ) ).

% in_entriesI
thf(fact_860_not__None__eq,axiom,
    ! [X: option3893003681250311888_d_nat] :
      ( ( X != none_P7808822419737304031_d_nat )
      = ( ? [Y4: product_prod_d_nat] :
            ( X
            = ( some_P734314052486210395_d_nat @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_861_not__None__eq,axiom,
    ! [X: option3723494939309271012_a_b_a] :
      ( ( X != none_P6894599056745261299_a_b_a )
      = ( ? [Y4: produc7822699449245016212_a_b_a] :
            ( X
            = ( some_P2288122801350569071_a_b_a @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_862_not__None__eq,axiom,
    ! [X: option5281194463001729774od_a_b] :
      ( ( X != none_P1914705788083608841od_a_b )
      = ( ? [Y4: product_prod_a_b] :
            ( X
            = ( some_P3592067299498605709od_a_b @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_863_not__None__eq,axiom,
    ! [X: option1943781222315457079_a_b_a] :
      ( ( X != none_m5415144799477956294_a_b_a )
      = ( ? [Y4: mappin2365780136655028071_a_b_a] :
            ( X
            = ( some_m5820982296856387138_a_b_a @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_864_not__None__eq,axiom,
    ! [X: option_b] :
      ( ( X != none_b )
      = ( ? [Y4: b] :
            ( X
            = ( some_b @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_865_not__None__eq,axiom,
    ! [X: option_a] :
      ( ( X != none_a )
      = ( ? [Y4: a] :
            ( X
            = ( some_a @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_866_not__Some__eq,axiom,
    ! [X: option3893003681250311888_d_nat] :
      ( ( ! [Y4: product_prod_d_nat] :
            ( X
           != ( some_P734314052486210395_d_nat @ Y4 ) ) )
      = ( X = none_P7808822419737304031_d_nat ) ) ).

% not_Some_eq
thf(fact_867_not__Some__eq,axiom,
    ! [X: option3723494939309271012_a_b_a] :
      ( ( ! [Y4: produc7822699449245016212_a_b_a] :
            ( X
           != ( some_P2288122801350569071_a_b_a @ Y4 ) ) )
      = ( X = none_P6894599056745261299_a_b_a ) ) ).

% not_Some_eq
thf(fact_868_not__Some__eq,axiom,
    ! [X: option5281194463001729774od_a_b] :
      ( ( ! [Y4: product_prod_a_b] :
            ( X
           != ( some_P3592067299498605709od_a_b @ Y4 ) ) )
      = ( X = none_P1914705788083608841od_a_b ) ) ).

% not_Some_eq
thf(fact_869_not__Some__eq,axiom,
    ! [X: option1943781222315457079_a_b_a] :
      ( ( ! [Y4: mappin2365780136655028071_a_b_a] :
            ( X
           != ( some_m5820982296856387138_a_b_a @ Y4 ) ) )
      = ( X = none_m5415144799477956294_a_b_a ) ) ).

% not_Some_eq
thf(fact_870_not__Some__eq,axiom,
    ! [X: option_b] :
      ( ( ! [Y4: b] :
            ( X
           != ( some_b @ Y4 ) ) )
      = ( X = none_b ) ) ).

% not_Some_eq
thf(fact_871_not__Some__eq,axiom,
    ! [X: option_a] :
      ( ( ! [Y4: a] :
            ( X
           != ( some_a @ Y4 ) ) )
      = ( X = none_a ) ) ).

% not_Some_eq
thf(fact_872_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_d_nat] :
      ( none_P7808822419737304031_d_nat
     != ( some_P734314052486210395_d_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_873_option_Odistinct_I1_J,axiom,
    ! [X2: produc7822699449245016212_a_b_a] :
      ( none_P6894599056745261299_a_b_a
     != ( some_P2288122801350569071_a_b_a @ X2 ) ) ).

% option.distinct(1)
thf(fact_874_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_a_b] :
      ( none_P1914705788083608841od_a_b
     != ( some_P3592067299498605709od_a_b @ X2 ) ) ).

% option.distinct(1)
thf(fact_875_option_Odistinct_I1_J,axiom,
    ! [X2: mappin2365780136655028071_a_b_a] :
      ( none_m5415144799477956294_a_b_a
     != ( some_m5820982296856387138_a_b_a @ X2 ) ) ).

% option.distinct(1)
thf(fact_876_option_Odistinct_I1_J,axiom,
    ! [X2: b] :
      ( none_b
     != ( some_b @ X2 ) ) ).

% option.distinct(1)
thf(fact_877_option_Odistinct_I1_J,axiom,
    ! [X2: a] :
      ( none_a
     != ( some_a @ X2 ) ) ).

% option.distinct(1)
thf(fact_878_option_OdiscI,axiom,
    ! [Option: option3893003681250311888_d_nat,X2: product_prod_d_nat] :
      ( ( Option
        = ( some_P734314052486210395_d_nat @ X2 ) )
     => ( Option != none_P7808822419737304031_d_nat ) ) ).

% option.discI
thf(fact_879_option_OdiscI,axiom,
    ! [Option: option3723494939309271012_a_b_a,X2: produc7822699449245016212_a_b_a] :
      ( ( Option
        = ( some_P2288122801350569071_a_b_a @ X2 ) )
     => ( Option != none_P6894599056745261299_a_b_a ) ) ).

% option.discI
thf(fact_880_option_OdiscI,axiom,
    ! [Option: option5281194463001729774od_a_b,X2: product_prod_a_b] :
      ( ( Option
        = ( some_P3592067299498605709od_a_b @ X2 ) )
     => ( Option != none_P1914705788083608841od_a_b ) ) ).

% option.discI
thf(fact_881_option_OdiscI,axiom,
    ! [Option: option1943781222315457079_a_b_a,X2: mappin2365780136655028071_a_b_a] :
      ( ( Option
        = ( some_m5820982296856387138_a_b_a @ X2 ) )
     => ( Option != none_m5415144799477956294_a_b_a ) ) ).

% option.discI
thf(fact_882_option_OdiscI,axiom,
    ! [Option: option_b,X2: b] :
      ( ( Option
        = ( some_b @ X2 ) )
     => ( Option != none_b ) ) ).

% option.discI
thf(fact_883_option_OdiscI,axiom,
    ! [Option: option_a,X2: a] :
      ( ( Option
        = ( some_a @ X2 ) )
     => ( Option != none_a ) ) ).

% option.discI
thf(fact_884_option_Oexhaust,axiom,
    ! [Y5: option3893003681250311888_d_nat] :
      ( ( Y5 != none_P7808822419737304031_d_nat )
     => ~ ! [X23: product_prod_d_nat] :
            ( Y5
           != ( some_P734314052486210395_d_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_885_option_Oexhaust,axiom,
    ! [Y5: option3723494939309271012_a_b_a] :
      ( ( Y5 != none_P6894599056745261299_a_b_a )
     => ~ ! [X23: produc7822699449245016212_a_b_a] :
            ( Y5
           != ( some_P2288122801350569071_a_b_a @ X23 ) ) ) ).

% option.exhaust
thf(fact_886_option_Oexhaust,axiom,
    ! [Y5: option5281194463001729774od_a_b] :
      ( ( Y5 != none_P1914705788083608841od_a_b )
     => ~ ! [X23: product_prod_a_b] :
            ( Y5
           != ( some_P3592067299498605709od_a_b @ X23 ) ) ) ).

% option.exhaust
thf(fact_887_option_Oexhaust,axiom,
    ! [Y5: option1943781222315457079_a_b_a] :
      ( ( Y5 != none_m5415144799477956294_a_b_a )
     => ~ ! [X23: mappin2365780136655028071_a_b_a] :
            ( Y5
           != ( some_m5820982296856387138_a_b_a @ X23 ) ) ) ).

% option.exhaust
thf(fact_888_option_Oexhaust,axiom,
    ! [Y5: option_b] :
      ( ( Y5 != none_b )
     => ~ ! [X23: b] :
            ( Y5
           != ( some_b @ X23 ) ) ) ).

% option.exhaust
thf(fact_889_option_Oexhaust,axiom,
    ! [Y5: option_a] :
      ( ( Y5 != none_a )
     => ~ ! [X23: a] :
            ( Y5
           != ( some_a @ X23 ) ) ) ).

% option.exhaust
thf(fact_890_split__option__ex,axiom,
    ( ( ^ [P8: option3893003681250311888_d_nat > $o] :
        ? [X7: option3893003681250311888_d_nat] : ( P8 @ X7 ) )
    = ( ^ [P5: option3893003681250311888_d_nat > $o] :
          ( ( P5 @ none_P7808822419737304031_d_nat )
          | ? [X4: product_prod_d_nat] : ( P5 @ ( some_P734314052486210395_d_nat @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_891_split__option__ex,axiom,
    ( ( ^ [P8: option3723494939309271012_a_b_a > $o] :
        ? [X7: option3723494939309271012_a_b_a] : ( P8 @ X7 ) )
    = ( ^ [P5: option3723494939309271012_a_b_a > $o] :
          ( ( P5 @ none_P6894599056745261299_a_b_a )
          | ? [X4: produc7822699449245016212_a_b_a] : ( P5 @ ( some_P2288122801350569071_a_b_a @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_892_split__option__ex,axiom,
    ( ( ^ [P8: option5281194463001729774od_a_b > $o] :
        ? [X7: option5281194463001729774od_a_b] : ( P8 @ X7 ) )
    = ( ^ [P5: option5281194463001729774od_a_b > $o] :
          ( ( P5 @ none_P1914705788083608841od_a_b )
          | ? [X4: product_prod_a_b] : ( P5 @ ( some_P3592067299498605709od_a_b @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_893_split__option__ex,axiom,
    ( ( ^ [P8: option1943781222315457079_a_b_a > $o] :
        ? [X7: option1943781222315457079_a_b_a] : ( P8 @ X7 ) )
    = ( ^ [P5: option1943781222315457079_a_b_a > $o] :
          ( ( P5 @ none_m5415144799477956294_a_b_a )
          | ? [X4: mappin2365780136655028071_a_b_a] : ( P5 @ ( some_m5820982296856387138_a_b_a @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_894_split__option__ex,axiom,
    ( ( ^ [P8: option_b > $o] :
        ? [X7: option_b] : ( P8 @ X7 ) )
    = ( ^ [P5: option_b > $o] :
          ( ( P5 @ none_b )
          | ? [X4: b] : ( P5 @ ( some_b @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_895_split__option__ex,axiom,
    ( ( ^ [P8: option_a > $o] :
        ? [X7: option_a] : ( P8 @ X7 ) )
    = ( ^ [P5: option_a > $o] :
          ( ( P5 @ none_a )
          | ? [X4: a] : ( P5 @ ( some_a @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_896_split__option__all,axiom,
    ( ( ^ [P8: option3893003681250311888_d_nat > $o] :
        ! [X7: option3893003681250311888_d_nat] : ( P8 @ X7 ) )
    = ( ^ [P5: option3893003681250311888_d_nat > $o] :
          ( ( P5 @ none_P7808822419737304031_d_nat )
          & ! [X4: product_prod_d_nat] : ( P5 @ ( some_P734314052486210395_d_nat @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_897_split__option__all,axiom,
    ( ( ^ [P8: option3723494939309271012_a_b_a > $o] :
        ! [X7: option3723494939309271012_a_b_a] : ( P8 @ X7 ) )
    = ( ^ [P5: option3723494939309271012_a_b_a > $o] :
          ( ( P5 @ none_P6894599056745261299_a_b_a )
          & ! [X4: produc7822699449245016212_a_b_a] : ( P5 @ ( some_P2288122801350569071_a_b_a @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_898_split__option__all,axiom,
    ( ( ^ [P8: option5281194463001729774od_a_b > $o] :
        ! [X7: option5281194463001729774od_a_b] : ( P8 @ X7 ) )
    = ( ^ [P5: option5281194463001729774od_a_b > $o] :
          ( ( P5 @ none_P1914705788083608841od_a_b )
          & ! [X4: product_prod_a_b] : ( P5 @ ( some_P3592067299498605709od_a_b @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_899_split__option__all,axiom,
    ( ( ^ [P8: option1943781222315457079_a_b_a > $o] :
        ! [X7: option1943781222315457079_a_b_a] : ( P8 @ X7 ) )
    = ( ^ [P5: option1943781222315457079_a_b_a > $o] :
          ( ( P5 @ none_m5415144799477956294_a_b_a )
          & ! [X4: mappin2365780136655028071_a_b_a] : ( P5 @ ( some_m5820982296856387138_a_b_a @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_900_split__option__all,axiom,
    ( ( ^ [P8: option_b > $o] :
        ! [X7: option_b] : ( P8 @ X7 ) )
    = ( ^ [P5: option_b > $o] :
          ( ( P5 @ none_b )
          & ! [X4: b] : ( P5 @ ( some_b @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_901_split__option__all,axiom,
    ( ( ^ [P8: option_a > $o] :
        ! [X7: option_a] : ( P8 @ X7 ) )
    = ( ^ [P5: option_a > $o] :
          ( ( P5 @ none_a )
          & ! [X4: a] : ( P5 @ ( some_a @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_902_combine__options__cases,axiom,
    ! [X: option_b,P: option_b > option_b > $o,Y5: option_b] :
      ( ( ( X = none_b )
       => ( P @ X @ Y5 ) )
     => ( ( ( Y5 = none_b )
         => ( P @ X @ Y5 ) )
       => ( ! [A3: b,B3: b] :
              ( ( X
                = ( some_b @ A3 ) )
             => ( ( Y5
                  = ( some_b @ B3 ) )
               => ( P @ X @ Y5 ) ) )
         => ( P @ X @ Y5 ) ) ) ) ).

% combine_options_cases
thf(fact_903_combine__options__cases,axiom,
    ! [X: option_b,P: option_b > option_a > $o,Y5: option_a] :
      ( ( ( X = none_b )
       => ( P @ X @ Y5 ) )
     => ( ( ( Y5 = none_a )
         => ( P @ X @ Y5 ) )
       => ( ! [A3: b,B3: a] :
              ( ( X
                = ( some_b @ A3 ) )
             => ( ( Y5
                  = ( some_a @ B3 ) )
               => ( P @ X @ Y5 ) ) )
         => ( P @ X @ Y5 ) ) ) ) ).

% combine_options_cases
thf(fact_904_combine__options__cases,axiom,
    ! [X: option_a,P: option_a > option_b > $o,Y5: option_b] :
      ( ( ( X = none_a )
       => ( P @ X @ Y5 ) )
     => ( ( ( Y5 = none_b )
         => ( P @ X @ Y5 ) )
       => ( ! [A3: a,B3: b] :
              ( ( X
                = ( some_a @ A3 ) )
             => ( ( Y5
                  = ( some_b @ B3 ) )
               => ( P @ X @ Y5 ) ) )
         => ( P @ X @ Y5 ) ) ) ) ).

% combine_options_cases
thf(fact_905_combine__options__cases,axiom,
    ! [X: option_a,P: option_a > option_a > $o,Y5: option_a] :
      ( ( ( X = none_a )
       => ( P @ X @ Y5 ) )
     => ( ( ( Y5 = none_a )
         => ( P @ X @ Y5 ) )
       => ( ! [A3: a,B3: a] :
              ( ( X
                = ( some_a @ A3 ) )
             => ( ( Y5
                  = ( some_a @ B3 ) )
               => ( P @ X @ Y5 ) ) )
         => ( P @ X @ Y5 ) ) ) ) ).

% combine_options_cases
thf(fact_906_combine__options__cases,axiom,
    ! [X: option3893003681250311888_d_nat,P: option3893003681250311888_d_nat > option_b > $o,Y5: option_b] :
      ( ( ( X = none_P7808822419737304031_d_nat )
       => ( P @ X @ Y5 ) )
     => ( ( ( Y5 = none_b )
         => ( P @ X @ Y5 ) )
       => ( ! [A3: product_prod_d_nat,B3: b] :
              ( ( X
                = ( some_P734314052486210395_d_nat @ A3 ) )
             => ( ( Y5
                  = ( some_b @ B3 ) )
               => ( P @ X @ Y5 ) ) )
         => ( P @ X @ Y5 ) ) ) ) ).

% combine_options_cases
thf(fact_907_combine__options__cases,axiom,
    ! [X: option3893003681250311888_d_nat,P: option3893003681250311888_d_nat > option_a > $o,Y5: option_a] :
      ( ( ( X = none_P7808822419737304031_d_nat )
       => ( P @ X @ Y5 ) )
     => ( ( ( Y5 = none_a )
         => ( P @ X @ Y5 ) )
       => ( ! [A3: product_prod_d_nat,B3: a] :
              ( ( X
                = ( some_P734314052486210395_d_nat @ A3 ) )
             => ( ( Y5
                  = ( some_a @ B3 ) )
               => ( P @ X @ Y5 ) ) )
         => ( P @ X @ Y5 ) ) ) ) ).

% combine_options_cases
thf(fact_908_combine__options__cases,axiom,
    ! [X: option5281194463001729774od_a_b,P: option5281194463001729774od_a_b > option_b > $o,Y5: option_b] :
      ( ( ( X = none_P1914705788083608841od_a_b )
       => ( P @ X @ Y5 ) )
     => ( ( ( Y5 = none_b )
         => ( P @ X @ Y5 ) )
       => ( ! [A3: product_prod_a_b,B3: b] :
              ( ( X
                = ( some_P3592067299498605709od_a_b @ A3 ) )
             => ( ( Y5
                  = ( some_b @ B3 ) )
               => ( P @ X @ Y5 ) ) )
         => ( P @ X @ Y5 ) ) ) ) ).

% combine_options_cases
thf(fact_909_combine__options__cases,axiom,
    ! [X: option5281194463001729774od_a_b,P: option5281194463001729774od_a_b > option_a > $o,Y5: option_a] :
      ( ( ( X = none_P1914705788083608841od_a_b )
       => ( P @ X @ Y5 ) )
     => ( ( ( Y5 = none_a )
         => ( P @ X @ Y5 ) )
       => ( ! [A3: product_prod_a_b,B3: a] :
              ( ( X
                = ( some_P3592067299498605709od_a_b @ A3 ) )
             => ( ( Y5
                  = ( some_a @ B3 ) )
               => ( P @ X @ Y5 ) ) )
         => ( P @ X @ Y5 ) ) ) ) ).

% combine_options_cases
thf(fact_910_combine__options__cases,axiom,
    ! [X: option_b,P: option_b > option3893003681250311888_d_nat > $o,Y5: option3893003681250311888_d_nat] :
      ( ( ( X = none_b )
       => ( P @ X @ Y5 ) )
     => ( ( ( Y5 = none_P7808822419737304031_d_nat )
         => ( P @ X @ Y5 ) )
       => ( ! [A3: b,B3: product_prod_d_nat] :
              ( ( X
                = ( some_b @ A3 ) )
             => ( ( Y5
                  = ( some_P734314052486210395_d_nat @ B3 ) )
               => ( P @ X @ Y5 ) ) )
         => ( P @ X @ Y5 ) ) ) ) ).

% combine_options_cases
thf(fact_911_combine__options__cases,axiom,
    ! [X: option_b,P: option_b > option5281194463001729774od_a_b > $o,Y5: option5281194463001729774od_a_b] :
      ( ( ( X = none_b )
       => ( P @ X @ Y5 ) )
     => ( ( ( Y5 = none_P1914705788083608841od_a_b )
         => ( P @ X @ Y5 ) )
       => ( ! [A3: b,B3: product_prod_a_b] :
              ( ( X
                = ( some_b @ A3 ) )
             => ( ( Y5
                  = ( some_P3592067299498605709od_a_b @ B3 ) )
               => ( P @ X @ Y5 ) ) )
         => ( P @ X @ Y5 ) ) ) ) ).

% combine_options_cases
thf(fact_912_option_Osimps_I4_J,axiom,
    ! [F1: produc7822699449245016212_a_b_a,F22: a > produc7822699449245016212_a_b_a] :
      ( ( case_o140846598303662877_b_a_a @ F1 @ F22 @ none_a )
      = F1 ) ).

% option.simps(4)
thf(fact_913_option_Osimps_I4_J,axiom,
    ! [F1: mappin2365780136655028071_a_b_a,F22: a > mappin2365780136655028071_a_b_a] :
      ( ( case_o2154183385986904842_b_a_a @ F1 @ F22 @ none_a )
      = F1 ) ).

% option.simps(4)
thf(fact_914_option_Osimps_I4_J,axiom,
    ! [F1: option3893003681250311888_d_nat,F22: product_prod_d_nat > option3893003681250311888_d_nat] :
      ( ( case_o6579539980594861021_d_nat @ F1 @ F22 @ none_P7808822419737304031_d_nat )
      = F1 ) ).

% option.simps(4)
thf(fact_915_option_Osimps_I4_J,axiom,
    ! [F1: option_a,F22: a > option_a] :
      ( ( case_o3148979394504432965on_a_a @ F1 @ F22 @ none_a )
      = F1 ) ).

% option.simps(4)
thf(fact_916_option_Osimps_I4_J,axiom,
    ! [F1: a,F22: a > a] :
      ( ( case_option_a_a @ F1 @ F22 @ none_a )
      = F1 ) ).

% option.simps(4)
thf(fact_917_option_Osimps_I4_J,axiom,
    ! [F1: $o,F22: a > $o] :
      ( ( case_option_o_a @ F1 @ F22 @ none_a )
      = F1 ) ).

% option.simps(4)
thf(fact_918_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option3893003681250311888_d_nat] :
      ( ( Option != none_P7808822419737304031_d_nat )
      = ( case_o4860870191544392141_d_nat @ $false
        @ ^ [Uu: product_prod_d_nat] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_919_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_a] :
      ( ( Option != none_a )
      = ( case_option_o_a @ $false
        @ ^ [Uu: a] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_920_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_a] :
      ( ( Option = none_a )
      = ( case_option_o_a @ $true
        @ ^ [Uu: a] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_921_case__optionE,axiom,
    ! [P: $o,Q2: a > $o,X: option_a] :
      ( ( case_option_o_a @ P @ Q2 @ X )
     => ( ( ( X = none_a )
         => ~ P )
       => ~ ! [Y3: a] :
              ( ( X
                = ( some_a @ Y3 ) )
             => ~ ( Q2 @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_922_listrel__Cons2,axiom,
    ! [Xs2: list_a,Y5: produc5456268773499131775_d_nat,Ys3: list_P7053655179267667343_d_nat,R3: set_Pr8583615357880026082_d_nat] :
      ( ( member7164185383468370905_d_nat @ ( produc6231069337089993148_d_nat @ Xs2 @ ( cons_P8522577077107056457_d_nat @ Y5 @ Ys3 ) ) @ ( listre7853655052829806284_d_nat @ R3 ) )
     => ~ ! [X3: a,Xs: list_a] :
            ( ( Xs2
              = ( cons_a @ X3 @ Xs ) )
           => ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X3 @ Y5 ) @ R3 )
             => ~ ( member7164185383468370905_d_nat @ ( produc6231069337089993148_d_nat @ Xs @ Ys3 ) @ ( listre7853655052829806284_d_nat @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_923_listrel__Cons2,axiom,
    ! [Xs2: list_a,Y5: b,Ys3: list_b,R3: set_Product_prod_a_b] :
      ( ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ Xs2 @ ( cons_b @ Y5 @ Ys3 ) ) @ ( listrel_a_b @ R3 ) )
     => ~ ! [X3: a,Xs: list_a] :
            ( ( Xs2
              = ( cons_a @ X3 @ Xs ) )
           => ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X3 @ Y5 ) @ R3 )
             => ~ ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ Xs @ Ys3 ) @ ( listrel_a_b @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_924_listrel__Cons2,axiom,
    ! [Xs2: list_P6684558232516339260_d_nat,Y5: mappin2365780136655028071_a_b_a,Ys3: list_m9210641325446214647_a_b_a,R3: set_Pr8226792695654363348_a_b_a] :
      ( ( member2359517100237508157_a_b_a @ ( produc6307639320882748236_a_b_a @ Xs2 @ ( cons_m8574943908289518641_a_b_a @ Y5 @ Ys3 ) ) @ ( listre3191202743213796614_a_b_a @ R3 ) )
     => ~ ! [X3: produc1990351791824763820_d_nat,Xs: list_P6684558232516339260_d_nat] :
            ( ( Xs2
              = ( cons_P2594832900118292854_d_nat @ X3 @ Xs ) )
           => ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X3 @ Y5 ) @ R3 )
             => ~ ( member2359517100237508157_a_b_a @ ( produc6307639320882748236_a_b_a @ Xs @ Ys3 ) @ ( listre3191202743213796614_a_b_a @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_925_listrel__Cons2,axiom,
    ! [Xs2: list_l337834041258660428_d_nat,Y5: mappin2365780136655028071_a_b_a,Ys3: list_m9210641325446214647_a_b_a,R3: set_Pr2043227878544749668_a_b_a] :
      ( ( member3296818671409482061_a_b_a @ ( produc3873751758850487132_a_b_a @ Xs2 @ ( cons_m8574943908289518641_a_b_a @ Y5 @ Ys3 ) ) @ ( listre6449073666664715926_a_b_a @ R3 ) )
     => ~ ! [X3: list_P6684558232516339260_d_nat,Xs: list_l337834041258660428_d_nat] :
            ( ( Xs2
              = ( cons_l4410424432949683974_d_nat @ X3 @ Xs ) )
           => ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X3 @ Y5 ) @ R3 )
             => ~ ( member3296818671409482061_a_b_a @ ( produc3873751758850487132_a_b_a @ Xs @ Ys3 ) @ ( listre6449073666664715926_a_b_a @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_926_listrel__Cons2,axiom,
    ! [Xs2: list_a,Y5: mappin2365780136655028071_a_b_a,Ys3: list_m9210641325446214647_a_b_a,R3: set_Pr2084289749507318346_a_b_a] :
      ( ( member5215947914154143169_a_b_a @ ( produc1462633373383792292_a_b_a @ Xs2 @ ( cons_m8574943908289518641_a_b_a @ Y5 @ Ys3 ) ) @ ( listre3604961095617705268_a_b_a @ R3 ) )
     => ~ ! [X3: a,Xs: list_a] :
            ( ( Xs2
              = ( cons_a @ X3 @ Xs ) )
           => ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X3 @ Y5 ) @ R3 )
             => ~ ( member5215947914154143169_a_b_a @ ( produc1462633373383792292_a_b_a @ Xs @ Ys3 ) @ ( listre3604961095617705268_a_b_a @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_927_listrel__Cons1,axiom,
    ! [Y5: a,Ys3: list_a,Xs2: list_P7053655179267667343_d_nat,R3: set_Pr8583615357880026082_d_nat] :
      ( ( member7164185383468370905_d_nat @ ( produc6231069337089993148_d_nat @ ( cons_a @ Y5 @ Ys3 ) @ Xs2 ) @ ( listre7853655052829806284_d_nat @ R3 ) )
     => ~ ! [Y3: produc5456268773499131775_d_nat,Ys: list_P7053655179267667343_d_nat] :
            ( ( Xs2
              = ( cons_P8522577077107056457_d_nat @ Y3 @ Ys ) )
           => ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ Y5 @ Y3 ) @ R3 )
             => ~ ( member7164185383468370905_d_nat @ ( produc6231069337089993148_d_nat @ Ys3 @ Ys ) @ ( listre7853655052829806284_d_nat @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_928_listrel__Cons1,axiom,
    ! [Y5: a,Ys3: list_a,Xs2: list_b,R3: set_Product_prod_a_b] :
      ( ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ ( cons_a @ Y5 @ Ys3 ) @ Xs2 ) @ ( listrel_a_b @ R3 ) )
     => ~ ! [Y3: b,Ys: list_b] :
            ( ( Xs2
              = ( cons_b @ Y3 @ Ys ) )
           => ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ Y5 @ Y3 ) @ R3 )
             => ~ ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ Ys3 @ Ys ) @ ( listrel_a_b @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_929_listrel__Cons1,axiom,
    ! [Y5: produc1990351791824763820_d_nat,Ys3: list_P6684558232516339260_d_nat,Xs2: list_m9210641325446214647_a_b_a,R3: set_Pr8226792695654363348_a_b_a] :
      ( ( member2359517100237508157_a_b_a @ ( produc6307639320882748236_a_b_a @ ( cons_P2594832900118292854_d_nat @ Y5 @ Ys3 ) @ Xs2 ) @ ( listre3191202743213796614_a_b_a @ R3 ) )
     => ~ ! [Y3: mappin2365780136655028071_a_b_a,Ys: list_m9210641325446214647_a_b_a] :
            ( ( Xs2
              = ( cons_m8574943908289518641_a_b_a @ Y3 @ Ys ) )
           => ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ Y5 @ Y3 ) @ R3 )
             => ~ ( member2359517100237508157_a_b_a @ ( produc6307639320882748236_a_b_a @ Ys3 @ Ys ) @ ( listre3191202743213796614_a_b_a @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_930_listrel__Cons1,axiom,
    ! [Y5: list_P6684558232516339260_d_nat,Ys3: list_l337834041258660428_d_nat,Xs2: list_m9210641325446214647_a_b_a,R3: set_Pr2043227878544749668_a_b_a] :
      ( ( member3296818671409482061_a_b_a @ ( produc3873751758850487132_a_b_a @ ( cons_l4410424432949683974_d_nat @ Y5 @ Ys3 ) @ Xs2 ) @ ( listre6449073666664715926_a_b_a @ R3 ) )
     => ~ ! [Y3: mappin2365780136655028071_a_b_a,Ys: list_m9210641325446214647_a_b_a] :
            ( ( Xs2
              = ( cons_m8574943908289518641_a_b_a @ Y3 @ Ys ) )
           => ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ Y5 @ Y3 ) @ R3 )
             => ~ ( member3296818671409482061_a_b_a @ ( produc3873751758850487132_a_b_a @ Ys3 @ Ys ) @ ( listre6449073666664715926_a_b_a @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_931_listrel__Cons1,axiom,
    ! [Y5: a,Ys3: list_a,Xs2: list_m9210641325446214647_a_b_a,R3: set_Pr2084289749507318346_a_b_a] :
      ( ( member5215947914154143169_a_b_a @ ( produc1462633373383792292_a_b_a @ ( cons_a @ Y5 @ Ys3 ) @ Xs2 ) @ ( listre3604961095617705268_a_b_a @ R3 ) )
     => ~ ! [Y3: mappin2365780136655028071_a_b_a,Ys: list_m9210641325446214647_a_b_a] :
            ( ( Xs2
              = ( cons_m8574943908289518641_a_b_a @ Y3 @ Ys ) )
           => ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ Y5 @ Y3 ) @ R3 )
             => ~ ( member5215947914154143169_a_b_a @ ( produc1462633373383792292_a_b_a @ Ys3 @ Ys ) @ ( listre3604961095617705268_a_b_a @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_932_listrel_OCons,axiom,
    ! [X: a,Y5: produc5456268773499131775_d_nat,R3: set_Pr8583615357880026082_d_nat,Xs2: list_a,Ys3: list_P7053655179267667343_d_nat] :
      ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X @ Y5 ) @ R3 )
     => ( ( member7164185383468370905_d_nat @ ( produc6231069337089993148_d_nat @ Xs2 @ Ys3 ) @ ( listre7853655052829806284_d_nat @ R3 ) )
       => ( member7164185383468370905_d_nat @ ( produc6231069337089993148_d_nat @ ( cons_a @ X @ Xs2 ) @ ( cons_P8522577077107056457_d_nat @ Y5 @ Ys3 ) ) @ ( listre7853655052829806284_d_nat @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_933_listrel_OCons,axiom,
    ! [X: a,Y5: b,R3: set_Product_prod_a_b,Xs2: list_a,Ys3: list_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X @ Y5 ) @ R3 )
     => ( ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ Xs2 @ Ys3 ) @ ( listrel_a_b @ R3 ) )
       => ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ ( cons_a @ X @ Xs2 ) @ ( cons_b @ Y5 @ Ys3 ) ) @ ( listrel_a_b @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_934_listrel_OCons,axiom,
    ! [X: produc1990351791824763820_d_nat,Y5: mappin2365780136655028071_a_b_a,R3: set_Pr8226792695654363348_a_b_a,Xs2: list_P6684558232516339260_d_nat,Ys3: list_m9210641325446214647_a_b_a] :
      ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X @ Y5 ) @ R3 )
     => ( ( member2359517100237508157_a_b_a @ ( produc6307639320882748236_a_b_a @ Xs2 @ Ys3 ) @ ( listre3191202743213796614_a_b_a @ R3 ) )
       => ( member2359517100237508157_a_b_a @ ( produc6307639320882748236_a_b_a @ ( cons_P2594832900118292854_d_nat @ X @ Xs2 ) @ ( cons_m8574943908289518641_a_b_a @ Y5 @ Ys3 ) ) @ ( listre3191202743213796614_a_b_a @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_935_listrel_OCons,axiom,
    ! [X: list_P6684558232516339260_d_nat,Y5: mappin2365780136655028071_a_b_a,R3: set_Pr2043227878544749668_a_b_a,Xs2: list_l337834041258660428_d_nat,Ys3: list_m9210641325446214647_a_b_a] :
      ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X @ Y5 ) @ R3 )
     => ( ( member3296818671409482061_a_b_a @ ( produc3873751758850487132_a_b_a @ Xs2 @ Ys3 ) @ ( listre6449073666664715926_a_b_a @ R3 ) )
       => ( member3296818671409482061_a_b_a @ ( produc3873751758850487132_a_b_a @ ( cons_l4410424432949683974_d_nat @ X @ Xs2 ) @ ( cons_m8574943908289518641_a_b_a @ Y5 @ Ys3 ) ) @ ( listre6449073666664715926_a_b_a @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_936_listrel_OCons,axiom,
    ! [X: a,Y5: mappin2365780136655028071_a_b_a,R3: set_Pr2084289749507318346_a_b_a,Xs2: list_a,Ys3: list_m9210641325446214647_a_b_a] :
      ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X @ Y5 ) @ R3 )
     => ( ( member5215947914154143169_a_b_a @ ( produc1462633373383792292_a_b_a @ Xs2 @ Ys3 ) @ ( listre3604961095617705268_a_b_a @ R3 ) )
       => ( member5215947914154143169_a_b_a @ ( produc1462633373383792292_a_b_a @ ( cons_a @ X @ Xs2 ) @ ( cons_m8574943908289518641_a_b_a @ Y5 @ Ys3 ) ) @ ( listre3604961095617705268_a_b_a @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_937_delete__aux_Osimps_I2_J,axiom,
    ! [K2: a,K3: a,V2: produc5456268773499131775_d_nat,Xs2: list_P6684558232516339260_d_nat] :
      ( ( ( K2 = K3 )
       => ( ( delete6174134448749622103_d_nat @ K2 @ ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K3 @ V2 ) @ Xs2 ) )
          = Xs2 ) )
      & ( ( K2 != K3 )
       => ( ( delete6174134448749622103_d_nat @ K2 @ ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K3 @ V2 ) @ Xs2 ) )
          = ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K3 @ V2 ) @ ( delete6174134448749622103_d_nat @ K2 @ Xs2 ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_938_delete__aux_Osimps_I2_J,axiom,
    ! [K2: a,K3: a,V2: b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( ( K2 = K3 )
       => ( ( delete_aux_a_b @ K2 @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V2 ) @ Xs2 ) )
          = Xs2 ) )
      & ( ( K2 != K3 )
       => ( ( delete_aux_a_b @ K2 @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V2 ) @ Xs2 ) )
          = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K3 @ V2 ) @ ( delete_aux_a_b @ K2 @ Xs2 ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_939_delete__aux_Osimps_I2_J,axiom,
    ! [K2: produc1990351791824763820_d_nat,K3: produc1990351791824763820_d_nat,V2: mappin2365780136655028071_a_b_a,Xs2: list_P4793483941206178810_a_b_a] :
      ( ( ( K2 = K3 )
       => ( ( delete7618755331649393595_a_b_a @ K2 @ ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K3 @ V2 ) @ Xs2 ) )
          = Xs2 ) )
      & ( ( K2 != K3 )
       => ( ( delete7618755331649393595_a_b_a @ K2 @ ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K3 @ V2 ) @ Xs2 ) )
          = ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K3 @ V2 ) @ ( delete7618755331649393595_a_b_a @ K2 @ Xs2 ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_940_delete__aux_Osimps_I2_J,axiom,
    ! [K2: list_P6684558232516339260_d_nat,K3: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a,Xs2: list_P7611574810425128074_a_b_a] :
      ( ( ( K2 = K3 )
       => ( ( delete3219952628631576267_a_b_a @ K2 @ ( cons_P2804281896780507450_a_b_a @ ( produc2616710729536263484_a_b_a @ K3 @ V2 ) @ Xs2 ) )
          = Xs2 ) )
      & ( ( K2 != K3 )
       => ( ( delete3219952628631576267_a_b_a @ K2 @ ( cons_P2804281896780507450_a_b_a @ ( produc2616710729536263484_a_b_a @ K3 @ V2 ) @ Xs2 ) )
          = ( cons_P2804281896780507450_a_b_a @ ( produc2616710729536263484_a_b_a @ K3 @ V2 ) @ ( delete3219952628631576267_a_b_a @ K2 @ Xs2 ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_941_delete__aux_Osimps_I2_J,axiom,
    ! [K2: a,K3: a,V2: mappin2365780136655028071_a_b_a,Xs2: list_P2992919015626629540_a_b_a] :
      ( ( ( K2 = K3 )
       => ( ( delete1755771892824606527_a_b_a @ K2 @ ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K3 @ V2 ) @ Xs2 ) )
          = Xs2 ) )
      & ( ( K2 != K3 )
       => ( ( delete1755771892824606527_a_b_a @ K2 @ ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K3 @ V2 ) @ Xs2 ) )
          = ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K3 @ V2 ) @ ( delete1755771892824606527_a_b_a @ K2 @ Xs2 ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_942_in__entriesD,axiom,
    ! [K2: product_prod_a_b,V2: a,M: mappin2365780136655028071_a_b_a] :
      ( ( member2194267517197600294_a_b_a @ ( produc5056887847982525421_a_b_a @ K2 @ V2 ) @ ( entrie935192368317430852_a_b_a @ M ) )
     => ( ( lookup863726980359885586_a_b_a @ M @ K2 )
        = ( some_a @ V2 ) ) ) ).

% in_entriesD
thf(fact_943_in__entriesD,axiom,
    ! [K2: a,V2: produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ K2 @ V2 ) @ ( entrie8866367667353307151_d_nat @ M ) )
     => ( ( lookup8480750236328838337_d_nat @ M @ K2 )
        = ( some_P3314015041498521818_d_nat @ V2 ) ) ) ).

% in_entriesD
thf(fact_944_in__entriesD,axiom,
    ! [K2: a,V2: b,M: mapping_a_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ ( entries_a_b @ M ) )
     => ( ( lookup_a_b @ M @ K2 )
        = ( some_b @ V2 ) ) ) ).

% in_entriesD
thf(fact_945_in__entriesD,axiom,
    ! [K2: produc1990351791824763820_d_nat,V2: mappin2365780136655028071_a_b_a,M: mappin1208405706615879902_a_b_a] :
      ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ K2 @ V2 ) @ ( entrie369595780118757635_a_b_a @ M ) )
     => ( ( lookup2444594617825954769_a_b_a @ M @ K2 )
        = ( some_m5820982296856387138_a_b_a @ V2 ) ) ) ).

% in_entriesD
thf(fact_946_in__entriesD,axiom,
    ! [K2: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ K2 @ V2 ) @ ( entrie4359402741916977171_a_b_a @ M ) )
     => ( ( lookup4661578797684577249_a_b_a @ M @ K2 )
        = ( some_m5820982296856387138_a_b_a @ V2 ) ) ) ).

% in_entriesD
thf(fact_947_in__entriesD,axiom,
    ! [K2: a,V2: mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ K2 @ V2 ) @ ( entrie7794251174888650743_a_b_a @ M ) )
     => ( ( lookup1261341372242361769_a_b_a @ M @ K2 )
        = ( some_m5820982296856387138_a_b_a @ V2 ) ) ) ).

% in_entriesD
thf(fact_948_delete__aux_Opelims,axiom,
    ! [X: a,Xa: list_P6684558232516339260_d_nat,Y5: list_P6684558232516339260_d_nat] :
      ( ( ( delete6174134448749622103_d_nat @ X @ Xa )
        = Y5 )
     => ( ( accp_P7714024643000875680_d_nat @ delete3420223133863848810_d_nat @ ( produc919746265332163299_d_nat @ X @ Xa ) )
       => ( ( ( Xa = nil_Pr2300798825815202086_d_nat )
           => ( ( Y5 = nil_Pr2300798825815202086_d_nat )
             => ~ ( accp_P7714024643000875680_d_nat @ delete3420223133863848810_d_nat @ ( produc919746265332163299_d_nat @ X @ nil_Pr2300798825815202086_d_nat ) ) ) )
         => ~ ! [K6: a,V4: produc5456268773499131775_d_nat,Xs: list_P6684558232516339260_d_nat] :
                ( ( Xa
                  = ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K6 @ V4 ) @ Xs ) )
               => ( ( ( ( X = K6 )
                     => ( Y5 = Xs ) )
                    & ( ( X != K6 )
                     => ( Y5
                        = ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K6 @ V4 ) @ ( delete6174134448749622103_d_nat @ X @ Xs ) ) ) ) )
                 => ~ ( accp_P7714024643000875680_d_nat @ delete3420223133863848810_d_nat @ ( produc919746265332163299_d_nat @ X @ ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K6 @ V4 ) @ Xs ) ) ) ) ) ) ) ) ).

% delete_aux.pelims
thf(fact_949_delete__aux_Opelims,axiom,
    ! [X: a,Xa: list_P1467974523212632366od_a_b,Y5: list_P1467974523212632366od_a_b] :
      ( ( ( delete_aux_a_b @ X @ Xa )
        = Y5 )
     => ( ( accp_P6845111368451614688od_a_b @ delete_aux_rel_a_b @ ( produc6643264353547226055od_a_b @ X @ Xa ) )
       => ( ( ( Xa = nil_Product_prod_a_b )
           => ( ( Y5 = nil_Product_prod_a_b )
             => ~ ( accp_P6845111368451614688od_a_b @ delete_aux_rel_a_b @ ( produc6643264353547226055od_a_b @ X @ nil_Product_prod_a_b ) ) ) )
         => ~ ! [K6: a,V4: b,Xs: list_P1467974523212632366od_a_b] :
                ( ( Xa
                  = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K6 @ V4 ) @ Xs ) )
               => ( ( ( ( X = K6 )
                     => ( Y5 = Xs ) )
                    & ( ( X != K6 )
                     => ( Y5
                        = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K6 @ V4 ) @ ( delete_aux_a_b @ X @ Xs ) ) ) ) )
                 => ~ ( accp_P6845111368451614688od_a_b @ delete_aux_rel_a_b @ ( produc6643264353547226055od_a_b @ X @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K6 @ V4 ) @ Xs ) ) ) ) ) ) ) ) ).

% delete_aux.pelims
thf(fact_950_delete__aux_Opelims,axiom,
    ! [X: produc1990351791824763820_d_nat,Xa: list_P4793483941206178810_a_b_a,Y5: list_P4793483941206178810_a_b_a] :
      ( ( ( delete7618755331649393595_a_b_a @ X @ Xa )
        = Y5 )
     => ( ( accp_P8454841393138718970_a_b_a @ delete6710275894556251368_a_b_a @ ( produc712968421101327221_a_b_a @ X @ Xa ) )
       => ( ( ( Xa = nil_Pr3763312433997567610_a_b_a )
           => ( ( Y5 = nil_Pr3763312433997567610_a_b_a )
             => ~ ( accp_P8454841393138718970_a_b_a @ delete6710275894556251368_a_b_a @ ( produc712968421101327221_a_b_a @ X @ nil_Pr3763312433997567610_a_b_a ) ) ) )
         => ~ ! [K6: produc1990351791824763820_d_nat,V4: mappin2365780136655028071_a_b_a,Xs: list_P4793483941206178810_a_b_a] :
                ( ( Xa
                  = ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K6 @ V4 ) @ Xs ) )
               => ( ( ( ( X = K6 )
                     => ( Y5 = Xs ) )
                    & ( ( X != K6 )
                     => ( Y5
                        = ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K6 @ V4 ) @ ( delete7618755331649393595_a_b_a @ X @ Xs ) ) ) ) )
                 => ~ ( accp_P8454841393138718970_a_b_a @ delete6710275894556251368_a_b_a @ ( produc712968421101327221_a_b_a @ X @ ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K6 @ V4 ) @ Xs ) ) ) ) ) ) ) ) ).

% delete_aux.pelims
thf(fact_951_delete__aux_Opelims,axiom,
    ! [X: list_P6684558232516339260_d_nat,Xa: list_P7611574810425128074_a_b_a,Y5: list_P7611574810425128074_a_b_a] :
      ( ( ( delete3219952628631576267_a_b_a @ X @ Xa )
        = Y5 )
     => ( ( accp_P6506438267451010426_a_b_a @ delete3337979896995227000_a_b_a @ ( produc8862864803594885109_a_b_a @ X @ Xa ) )
       => ( ( ( Xa = nil_Pr1908111780246848906_a_b_a )
           => ( ( Y5 = nil_Pr1908111780246848906_a_b_a )
             => ~ ( accp_P6506438267451010426_a_b_a @ delete3337979896995227000_a_b_a @ ( produc8862864803594885109_a_b_a @ X @ nil_Pr1908111780246848906_a_b_a ) ) ) )
         => ~ ! [K6: list_P6684558232516339260_d_nat,V4: mappin2365780136655028071_a_b_a,Xs: list_P7611574810425128074_a_b_a] :
                ( ( Xa
                  = ( cons_P2804281896780507450_a_b_a @ ( produc2616710729536263484_a_b_a @ K6 @ V4 ) @ Xs ) )
               => ( ( ( ( X = K6 )
                     => ( Y5 = Xs ) )
                    & ( ( X != K6 )
                     => ( Y5
                        = ( cons_P2804281896780507450_a_b_a @ ( produc2616710729536263484_a_b_a @ K6 @ V4 ) @ ( delete3219952628631576267_a_b_a @ X @ Xs ) ) ) ) )
                 => ~ ( accp_P6506438267451010426_a_b_a @ delete3337979896995227000_a_b_a @ ( produc8862864803594885109_a_b_a @ X @ ( cons_P2804281896780507450_a_b_a @ ( produc2616710729536263484_a_b_a @ K6 @ V4 ) @ Xs ) ) ) ) ) ) ) ) ).

% delete_aux.pelims
thf(fact_952_delete__aux_Opelims,axiom,
    ! [X: a,Xa: list_P2992919015626629540_a_b_a,Y5: list_P2992919015626629540_a_b_a] :
      ( ( ( delete1755771892824606527_a_b_a @ X @ Xa )
        = Y5 )
     => ( ( accp_P209492061542291080_a_b_a @ delete9131875200943534290_a_b_a @ ( produc6994869176599099851_a_b_a @ X @ Xa ) )
       => ( ( ( Xa = nil_Pr7632893056509733646_a_b_a )
           => ( ( Y5 = nil_Pr7632893056509733646_a_b_a )
             => ~ ( accp_P209492061542291080_a_b_a @ delete9131875200943534290_a_b_a @ ( produc6994869176599099851_a_b_a @ X @ nil_Pr7632893056509733646_a_b_a ) ) ) )
         => ~ ! [K6: a,V4: mappin2365780136655028071_a_b_a,Xs: list_P2992919015626629540_a_b_a] :
                ( ( Xa
                  = ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K6 @ V4 ) @ Xs ) )
               => ( ( ( ( X = K6 )
                     => ( Y5 = Xs ) )
                    & ( ( X != K6 )
                     => ( Y5
                        = ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K6 @ V4 ) @ ( delete1755771892824606527_a_b_a @ X @ Xs ) ) ) ) )
                 => ~ ( accp_P209492061542291080_a_b_a @ delete9131875200943534290_a_b_a @ ( produc6994869176599099851_a_b_a @ X @ ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K6 @ V4 ) @ Xs ) ) ) ) ) ) ) ) ).

% delete_aux.pelims
thf(fact_953_lookup__filter,axiom,
    ! [P: product_prod_a_b > a > $o,M: mappin2365780136655028071_a_b_a,K2: product_prod_a_b] :
      ( ( lookup863726980359885586_a_b_a @ ( filter4555623327470326256_a_b_a @ P @ M ) @ K2 )
      = ( case_o3148979394504432965on_a_a @ none_a
        @ ^ [V: a] : ( if_option_a @ ( P @ K2 @ V ) @ ( some_a @ V ) @ none_a )
        @ ( lookup863726980359885586_a_b_a @ M @ K2 ) ) ) ).

% lookup_filter
thf(fact_954_listrelp__listrel__eq,axiom,
    ! [R3: set_Pr8583615357880026082_d_nat] :
      ( ( listre3688113617497796606_d_nat
        @ ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ R3 ) )
      = ( ^ [X4: list_a,Y4: list_P7053655179267667343_d_nat] : ( member7164185383468370905_d_nat @ ( produc6231069337089993148_d_nat @ X4 @ Y4 ) @ ( listre7853655052829806284_d_nat @ R3 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_955_listrelp__listrel__eq,axiom,
    ! [R3: set_Product_prod_a_b] :
      ( ( listrelp_a_b
        @ ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ R3 ) )
      = ( ^ [X4: list_a,Y4: list_b] : ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ X4 @ Y4 ) @ ( listrel_a_b @ R3 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_956_listrelp__listrel__eq,axiom,
    ! [R3: set_Pr8226792695654363348_a_b_a] :
      ( ( listre3385509792418263892_a_b_a
        @ ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ R3 ) )
      = ( ^ [X4: list_P6684558232516339260_d_nat,Y4: list_m9210641325446214647_a_b_a] : ( member2359517100237508157_a_b_a @ ( produc6307639320882748236_a_b_a @ X4 @ Y4 ) @ ( listre3191202743213796614_a_b_a @ R3 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_957_listrelp__listrel__eq,axiom,
    ! [R3: set_Pr2043227878544749668_a_b_a] :
      ( ( listre9005319261410926052_a_b_a
        @ ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ R3 ) )
      = ( ^ [X4: list_l337834041258660428_d_nat,Y4: list_m9210641325446214647_a_b_a] : ( member3296818671409482061_a_b_a @ ( produc3873751758850487132_a_b_a @ X4 @ Y4 ) @ ( listre6449073666664715926_a_b_a @ R3 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_958_listrelp__listrel__eq,axiom,
    ! [R3: set_Pr2084289749507318346_a_b_a] :
      ( ( listre7373438637330161510_a_b_a
        @ ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ R3 ) )
      = ( ^ [X4: list_a,Y4: list_m9210641325446214647_a_b_a] : ( member5215947914154143169_a_b_a @ ( produc1462633373383792292_a_b_a @ X4 @ Y4 ) @ ( listre3604961095617705268_a_b_a @ R3 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_959_listrel__def,axiom,
    ( listre7853655052829806284_d_nat
    = ( ^ [R2: set_Pr8583615357880026082_d_nat] :
          ( collec7310168451725883927_d_nat
          @ ( produc7570592611891919727_nat_o
            @ ( listre3688113617497796606_d_nat
              @ ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ R2 ) ) ) ) ) ) ).

% listrel_def
thf(fact_960_listrel__def,axiom,
    ( listrel_a_b
    = ( ^ [R2: set_Product_prod_a_b] :
          ( collec1014089183935468435list_b
          @ ( produc5384923214849584043st_b_o
            @ ( listrelp_a_b
              @ ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ R2 ) ) ) ) ) ) ).

% listrel_def
thf(fact_961_listrel__def,axiom,
    ( listre3191202743213796614_a_b_a
    = ( ^ [R2: set_Pr8226792695654363348_a_b_a] :
          ( collec5876475893036432511_a_b_a
          @ ( produc6030957657069740535_b_a_o
            @ ( listre3385509792418263892_a_b_a
              @ ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ R2 ) ) ) ) ) ) ).

% listrel_def
thf(fact_962_listrel__def,axiom,
    ( listre6449073666664715926_a_b_a
    = ( ^ [R2: set_Pr2043227878544749668_a_b_a] :
          ( collec2086272073679187599_a_b_a
          @ ( produc7316857958416314343_b_a_o
            @ ( listre9005319261410926052_a_b_a
              @ ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ R2 ) ) ) ) ) ) ).

% listrel_def
thf(fact_963_listrel__def,axiom,
    ( listre3604961095617705268_a_b_a
    = ( ^ [R2: set_Pr2084289749507318346_a_b_a] :
          ( collec1653772041195981567_a_b_a
          @ ( produc9056601799352565383_b_a_o
            @ ( listre7373438637330161510_a_b_a
              @ ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ R2 ) ) ) ) ) ) ).

% listrel_def
thf(fact_964_lookup_Orep__eq,axiom,
    lookup863726980359885586_a_b_a = rep_Pr1676580395783003895_a_b_a ).

% lookup.rep_eq
thf(fact_965_lookup_Oabs__eq,axiom,
    ! [Y5: product_prod_a_b > option_a] :
      ( ( lookup863726980359885586_a_b_a @ ( mappin7275691486367225288_a_b_a @ Y5 ) )
      = Y5 ) ).

% lookup.abs_eq
thf(fact_966_map__ran__simps_I2_J,axiom,
    ! [F: a > b > b,K2: a,V2: b,Ps: list_P1467974523212632366od_a_b] :
      ( ( map_ran_a_b_b @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ Ps ) )
      = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ ( F @ K2 @ V2 ) ) @ ( map_ran_a_b_b @ F @ Ps ) ) ) ).

% map_ran_simps(2)
thf(fact_967_map__ran__simps_I2_J,axiom,
    ! [F: a > b > mappin2365780136655028071_a_b_a,K2: a,V2: b,Ps: list_P1467974523212632366od_a_b] :
      ( ( map_ra84476893876701727_a_b_a @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ Ps ) )
      = ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K2 @ ( F @ K2 @ V2 ) ) @ ( map_ra84476893876701727_a_b_a @ F @ Ps ) ) ) ).

% map_ran_simps(2)
thf(fact_968_map__ran__simps_I2_J,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > b,K2: a,V2: mappin2365780136655028071_a_b_a,Ps: list_P2992919015626629540_a_b_a] :
      ( ( map_ra3413412011744336911_b_a_b @ F @ ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K2 @ V2 ) @ Ps ) )
      = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ ( F @ K2 @ V2 ) ) @ ( map_ra3413412011744336911_b_a_b @ F @ Ps ) ) ) ).

% map_ran_simps(2)
thf(fact_969_map__ran__simps_I2_J,axiom,
    ! [F: a > produc5456268773499131775_d_nat > b,K2: a,V2: produc5456268773499131775_d_nat,Ps: list_P6684558232516339260_d_nat] :
      ( ( map_ra6437063042551688695_nat_b @ F @ ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K2 @ V2 ) @ Ps ) )
      = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ ( F @ K2 @ V2 ) ) @ ( map_ra6437063042551688695_nat_b @ F @ Ps ) ) ) ).

% map_ran_simps(2)
thf(fact_970_map__ran__simps_I2_J,axiom,
    ! [F: a > b > produc5456268773499131775_d_nat,K2: a,V2: b,Ps: list_P1467974523212632366od_a_b] :
      ( ( map_ra5507442408803108407_d_nat @ F @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ Ps ) )
      = ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K2 @ ( F @ K2 @ V2 ) ) @ ( map_ra5507442408803108407_d_nat @ F @ Ps ) ) ) ).

% map_ran_simps(2)
thf(fact_971_map__ran__simps_I2_J,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,K2: a,V2: mappin2365780136655028071_a_b_a,Ps: list_P2992919015626629540_a_b_a] :
      ( ( map_ra8693790637767086679_a_b_a @ F @ ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K2 @ V2 ) @ Ps ) )
      = ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K2 @ ( F @ K2 @ V2 ) ) @ ( map_ra8693790637767086679_a_b_a @ F @ Ps ) ) ) ).

% map_ran_simps(2)
thf(fact_972_map__ran__simps_I2_J,axiom,
    ! [F: a > produc5456268773499131775_d_nat > mappin2365780136655028071_a_b_a,K2: a,V2: produc5456268773499131775_d_nat,Ps: list_P6684558232516339260_d_nat] :
      ( ( map_ra2832949629095358063_a_b_a @ F @ ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K2 @ V2 ) @ Ps ) )
      = ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K2 @ ( F @ K2 @ V2 ) ) @ ( map_ra2832949629095358063_a_b_a @ F @ Ps ) ) ) ).

% map_ran_simps(2)
thf(fact_973_map__ran__simps_I2_J,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a > produc5456268773499131775_d_nat,K2: a,V2: mappin2365780136655028071_a_b_a,Ps: list_P2992919015626629540_a_b_a] :
      ( ( map_ra5118305792500478575_d_nat @ F @ ( cons_P5314874490607035230_a_b_a @ ( produc1583556749610718734_a_b_a @ K2 @ V2 ) @ Ps ) )
      = ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K2 @ ( F @ K2 @ V2 ) ) @ ( map_ra5118305792500478575_d_nat @ F @ Ps ) ) ) ).

% map_ran_simps(2)
thf(fact_974_map__ran__simps_I2_J,axiom,
    ! [F: a > produc5456268773499131775_d_nat > produc5456268773499131775_d_nat,K2: a,V2: produc5456268773499131775_d_nat,Ps: list_P6684558232516339260_d_nat] :
      ( ( map_ra9806137992921735_d_nat @ F @ ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K2 @ V2 ) @ Ps ) )
      = ( cons_P2594832900118292854_d_nat @ ( produc5865182755557626534_d_nat @ K2 @ ( F @ K2 @ V2 ) ) @ ( map_ra9806137992921735_d_nat @ F @ Ps ) ) ) ).

% map_ran_simps(2)
thf(fact_975_map__ran__simps_I2_J,axiom,
    ! [F: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,K2: produc1990351791824763820_d_nat,V2: mappin2365780136655028071_a_b_a,Ps: list_P4793483941206178810_a_b_a] :
      ( ( map_ra143964223479814323_a_b_a @ F @ ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K2 @ V2 ) @ Ps ) )
      = ( cons_P673726611873665578_a_b_a @ ( produc1681437058057156268_a_b_a @ K2 @ ( F @ K2 @ V2 ) ) @ ( map_ra143964223479814323_a_b_a @ F @ Ps ) ) ) ).

% map_ran_simps(2)
thf(fact_976_reaches__onD,axiom,
    ! [Run: a > option3982279322286820476_d_nat,S3: a,T: c,B: option3893003681250311888_d_nat,Vs: list_P7053655179267667343_d_nat,S4: a] :
      ( ( reache5148256237901502841_d_nat @ Run @ S3 @ ( cons_P8522577077107056457_d_nat @ ( produc5462365204240479225_d_nat @ T @ B ) @ Vs ) @ S4 )
     => ? [S5: a] :
          ( ( ( Run @ S3 )
            = ( some_P4227418909054022151_d_nat @ ( produc5865182755557626534_d_nat @ S5 @ ( produc5462365204240479225_d_nat @ T @ B ) ) ) )
          & ( reache5148256237901502841_d_nat @ Run @ S5 @ Vs @ S4 ) ) ) ).

% reaches_onD
thf(fact_977_lookup__delete,axiom,
    ! [K2: product_prod_a_b,M: mappin2365780136655028071_a_b_a] :
      ( ( lookup863726980359885586_a_b_a @ ( delete7952242444366740995_a_b_a @ K2 @ M ) @ K2 )
      = none_a ) ).

% lookup_delete
thf(fact_978_lookup__delete__neq,axiom,
    ! [K2: product_prod_a_b,K3: product_prod_a_b,M: mappin2365780136655028071_a_b_a] :
      ( ( K2 != K3 )
     => ( ( lookup863726980359885586_a_b_a @ ( delete7952242444366740995_a_b_a @ K2 @ M ) @ K3 )
        = ( lookup863726980359885586_a_b_a @ M @ K3 ) ) ) ).

% lookup_delete_neq
thf(fact_979_reaches__on_Ointros_I2_J,axiom,
    ! [Run: a > option3982279322286820476_d_nat,S3: a,S4: a,V2: produc5456268773499131775_d_nat,Vs: list_P7053655179267667343_d_nat,S6: a] :
      ( ( ( Run @ S3 )
        = ( some_P4227418909054022151_d_nat @ ( produc5865182755557626534_d_nat @ S4 @ V2 ) ) )
     => ( ( reache5148256237901502841_d_nat @ Run @ S4 @ Vs @ S6 )
       => ( reache5148256237901502841_d_nat @ Run @ S3 @ ( cons_P8522577077107056457_d_nat @ V2 @ Vs ) @ S6 ) ) ) ).

% reaches_on.intros(2)
thf(fact_980_reaches__on_Ointros_I2_J,axiom,
    ! [Run: a > option5281194463001729774od_a_b,S3: a,S4: a,V2: b,Vs: list_b,S6: a] :
      ( ( ( Run @ S3 )
        = ( some_P3592067299498605709od_a_b @ ( product_Pair_a_b @ S4 @ V2 ) ) )
     => ( ( reaches_on_a_b @ Run @ S4 @ Vs @ S6 )
       => ( reaches_on_a_b @ Run @ S3 @ ( cons_b @ V2 @ Vs ) @ S6 ) ) ) ).

% reaches_on.intros(2)
thf(fact_981_reaches__on_Ointros_I2_J,axiom,
    ! [Run: produc1990351791824763820_d_nat > option2221477324614387130_a_b_a,S3: produc1990351791824763820_d_nat,S4: produc1990351791824763820_d_nat,V2: mappin2365780136655028071_a_b_a,Vs: list_m9210641325446214647_a_b_a,S6: produc1990351791824763820_d_nat] :
      ( ( ( Run @ S3 )
        = ( some_P6648320236558503769_a_b_a @ ( produc1681437058057156268_a_b_a @ S4 @ V2 ) ) )
     => ( ( reache6877781913633749273_a_b_a @ Run @ S4 @ Vs @ S6 )
       => ( reache6877781913633749273_a_b_a @ Run @ S3 @ ( cons_m8574943908289518641_a_b_a @ V2 @ Vs ) @ S6 ) ) ) ).

% reaches_on.intros(2)
thf(fact_982_reaches__on_Ointros_I2_J,axiom,
    ! [Run: list_P6684558232516339260_d_nat > option4722335503605243466_a_b_a,S3: list_P6684558232516339260_d_nat,S4: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a,Vs: list_m9210641325446214647_a_b_a,S6: list_P6684558232516339260_d_nat] :
      ( ( ( Run @ S3 )
        = ( some_P4038468990576657129_a_b_a @ ( produc2616710729536263484_a_b_a @ S4 @ V2 ) ) )
     => ( ( reache4265786078874959145_a_b_a @ Run @ S4 @ Vs @ S6 )
       => ( reache4265786078874959145_a_b_a @ Run @ S3 @ ( cons_m8574943908289518641_a_b_a @ V2 @ Vs ) @ S6 ) ) ) ).

% reaches_on.intros(2)
thf(fact_983_reaches__on_Ointros_I2_J,axiom,
    ! [Run: a > option3723494939309271012_a_b_a,S3: a,S4: a,V2: mappin2365780136655028071_a_b_a,Vs: list_m9210641325446214647_a_b_a,S6: a] :
      ( ( ( Run @ S3 )
        = ( some_P2288122801350569071_a_b_a @ ( produc1583556749610718734_a_b_a @ S4 @ V2 ) ) )
     => ( ( reache8162303588387466849_a_b_a @ Run @ S4 @ Vs @ S6 )
       => ( reache8162303588387466849_a_b_a @ Run @ S3 @ ( cons_m8574943908289518641_a_b_a @ V2 @ Vs ) @ S6 ) ) ) ).

% reaches_on.intros(2)
thf(fact_984_reaches__on__ConsD,axiom,
    ! [Run: a > option3982279322286820476_d_nat,Q: a,X: produc5456268773499131775_d_nat,Xs2: list_P7053655179267667343_d_nat,Q6: a] :
      ( ( reache5148256237901502841_d_nat @ Run @ Q @ ( cons_P8522577077107056457_d_nat @ X @ Xs2 ) @ Q6 )
     => ? [Q7: a] :
          ( ( ( Run @ Q )
            = ( some_P4227418909054022151_d_nat @ ( produc5865182755557626534_d_nat @ Q7 @ X ) ) )
          & ( reache5148256237901502841_d_nat @ Run @ Q7 @ Xs2 @ Q6 ) ) ) ).

% reaches_on_ConsD
thf(fact_985_reaches__on__ConsD,axiom,
    ! [Run: a > option5281194463001729774od_a_b,Q: a,X: b,Xs2: list_b,Q6: a] :
      ( ( reaches_on_a_b @ Run @ Q @ ( cons_b @ X @ Xs2 ) @ Q6 )
     => ? [Q7: a] :
          ( ( ( Run @ Q )
            = ( some_P3592067299498605709od_a_b @ ( product_Pair_a_b @ Q7 @ X ) ) )
          & ( reaches_on_a_b @ Run @ Q7 @ Xs2 @ Q6 ) ) ) ).

% reaches_on_ConsD
thf(fact_986_reaches__on__ConsD,axiom,
    ! [Run: produc1990351791824763820_d_nat > option2221477324614387130_a_b_a,Q: produc1990351791824763820_d_nat,X: mappin2365780136655028071_a_b_a,Xs2: list_m9210641325446214647_a_b_a,Q6: produc1990351791824763820_d_nat] :
      ( ( reache6877781913633749273_a_b_a @ Run @ Q @ ( cons_m8574943908289518641_a_b_a @ X @ Xs2 ) @ Q6 )
     => ? [Q7: produc1990351791824763820_d_nat] :
          ( ( ( Run @ Q )
            = ( some_P6648320236558503769_a_b_a @ ( produc1681437058057156268_a_b_a @ Q7 @ X ) ) )
          & ( reache6877781913633749273_a_b_a @ Run @ Q7 @ Xs2 @ Q6 ) ) ) ).

% reaches_on_ConsD
thf(fact_987_reaches__on__ConsD,axiom,
    ! [Run: list_P6684558232516339260_d_nat > option4722335503605243466_a_b_a,Q: list_P6684558232516339260_d_nat,X: mappin2365780136655028071_a_b_a,Xs2: list_m9210641325446214647_a_b_a,Q6: list_P6684558232516339260_d_nat] :
      ( ( reache4265786078874959145_a_b_a @ Run @ Q @ ( cons_m8574943908289518641_a_b_a @ X @ Xs2 ) @ Q6 )
     => ? [Q7: list_P6684558232516339260_d_nat] :
          ( ( ( Run @ Q )
            = ( some_P4038468990576657129_a_b_a @ ( produc2616710729536263484_a_b_a @ Q7 @ X ) ) )
          & ( reache4265786078874959145_a_b_a @ Run @ Q7 @ Xs2 @ Q6 ) ) ) ).

% reaches_on_ConsD
thf(fact_988_reaches__on__ConsD,axiom,
    ! [Run: a > option3723494939309271012_a_b_a,Q: a,X: mappin2365780136655028071_a_b_a,Xs2: list_m9210641325446214647_a_b_a,Q6: a] :
      ( ( reache8162303588387466849_a_b_a @ Run @ Q @ ( cons_m8574943908289518641_a_b_a @ X @ Xs2 ) @ Q6 )
     => ? [Q7: a] :
          ( ( ( Run @ Q )
            = ( some_P2288122801350569071_a_b_a @ ( produc1583556749610718734_a_b_a @ Q7 @ X ) ) )
          & ( reache8162303588387466849_a_b_a @ Run @ Q7 @ Xs2 @ Q6 ) ) ) ).

% reaches_on_ConsD
thf(fact_989_reaches__on_Ocases,axiom,
    ! [Run: a > option3982279322286820476_d_nat,A1: a,A22: list_P7053655179267667343_d_nat,A32: a] :
      ( ( reache5148256237901502841_d_nat @ Run @ A1 @ A22 @ A32 )
     => ( ( ( A22 = nil_Pr3001256489040166649_d_nat )
         => ( A32 != A1 ) )
       => ~ ! [S7: a,V4: produc5456268773499131775_d_nat,Vs2: list_P7053655179267667343_d_nat] :
              ( ( A22
                = ( cons_P8522577077107056457_d_nat @ V4 @ Vs2 ) )
             => ( ( ( Run @ A1 )
                  = ( some_P4227418909054022151_d_nat @ ( produc5865182755557626534_d_nat @ S7 @ V4 ) ) )
               => ~ ( reache5148256237901502841_d_nat @ Run @ S7 @ Vs2 @ A32 ) ) ) ) ) ).

% reaches_on.cases
thf(fact_990_reaches__on_Ocases,axiom,
    ! [Run: a > option5281194463001729774od_a_b,A1: a,A22: list_b,A32: a] :
      ( ( reaches_on_a_b @ Run @ A1 @ A22 @ A32 )
     => ( ( ( A22 = nil_b )
         => ( A32 != A1 ) )
       => ~ ! [S7: a,V4: b,Vs2: list_b] :
              ( ( A22
                = ( cons_b @ V4 @ Vs2 ) )
             => ( ( ( Run @ A1 )
                  = ( some_P3592067299498605709od_a_b @ ( product_Pair_a_b @ S7 @ V4 ) ) )
               => ~ ( reaches_on_a_b @ Run @ S7 @ Vs2 @ A32 ) ) ) ) ) ).

% reaches_on.cases
thf(fact_991_reaches__on_Ocases,axiom,
    ! [Run: produc1990351791824763820_d_nat > option2221477324614387130_a_b_a,A1: produc1990351791824763820_d_nat,A22: list_m9210641325446214647_a_b_a,A32: produc1990351791824763820_d_nat] :
      ( ( reache6877781913633749273_a_b_a @ Run @ A1 @ A22 @ A32 )
     => ( ( ( A22 = nil_ma546712388551201761_a_b_a )
         => ( A32 != A1 ) )
       => ~ ! [S7: produc1990351791824763820_d_nat,V4: mappin2365780136655028071_a_b_a,Vs2: list_m9210641325446214647_a_b_a] :
              ( ( A22
                = ( cons_m8574943908289518641_a_b_a @ V4 @ Vs2 ) )
             => ( ( ( Run @ A1 )
                  = ( some_P6648320236558503769_a_b_a @ ( produc1681437058057156268_a_b_a @ S7 @ V4 ) ) )
               => ~ ( reache6877781913633749273_a_b_a @ Run @ S7 @ Vs2 @ A32 ) ) ) ) ) ).

% reaches_on.cases
thf(fact_992_reaches__on_Ocases,axiom,
    ! [Run: list_P6684558232516339260_d_nat > option4722335503605243466_a_b_a,A1: list_P6684558232516339260_d_nat,A22: list_m9210641325446214647_a_b_a,A32: list_P6684558232516339260_d_nat] :
      ( ( reache4265786078874959145_a_b_a @ Run @ A1 @ A22 @ A32 )
     => ( ( ( A22 = nil_ma546712388551201761_a_b_a )
         => ( A32 != A1 ) )
       => ~ ! [S7: list_P6684558232516339260_d_nat,V4: mappin2365780136655028071_a_b_a,Vs2: list_m9210641325446214647_a_b_a] :
              ( ( A22
                = ( cons_m8574943908289518641_a_b_a @ V4 @ Vs2 ) )
             => ( ( ( Run @ A1 )
                  = ( some_P4038468990576657129_a_b_a @ ( produc2616710729536263484_a_b_a @ S7 @ V4 ) ) )
               => ~ ( reache4265786078874959145_a_b_a @ Run @ S7 @ Vs2 @ A32 ) ) ) ) ) ).

% reaches_on.cases
thf(fact_993_reaches__on_Ocases,axiom,
    ! [Run: a > option3723494939309271012_a_b_a,A1: a,A22: list_m9210641325446214647_a_b_a,A32: a] :
      ( ( reache8162303588387466849_a_b_a @ Run @ A1 @ A22 @ A32 )
     => ( ( ( A22 = nil_ma546712388551201761_a_b_a )
         => ( A32 != A1 ) )
       => ~ ! [S7: a,V4: mappin2365780136655028071_a_b_a,Vs2: list_m9210641325446214647_a_b_a] :
              ( ( A22
                = ( cons_m8574943908289518641_a_b_a @ V4 @ Vs2 ) )
             => ( ( ( Run @ A1 )
                  = ( some_P2288122801350569071_a_b_a @ ( produc1583556749610718734_a_b_a @ S7 @ V4 ) ) )
               => ~ ( reache8162303588387466849_a_b_a @ Run @ S7 @ Vs2 @ A32 ) ) ) ) ) ).

% reaches_on.cases
thf(fact_994_reaches__on_Osimps,axiom,
    ( reache5148256237901502841_d_nat
    = ( ^ [Run2: a > option3982279322286820476_d_nat,A12: a,A23: list_P7053655179267667343_d_nat,A33: a] :
          ( ? [S2: a] :
              ( ( A12 = S2 )
              & ( A23 = nil_Pr3001256489040166649_d_nat )
              & ( A33 = S2 ) )
          | ? [S2: a,S8: a,V: produc5456268773499131775_d_nat,Vs3: list_P7053655179267667343_d_nat,S9: a] :
              ( ( A12 = S2 )
              & ( A23
                = ( cons_P8522577077107056457_d_nat @ V @ Vs3 ) )
              & ( A33 = S9 )
              & ( ( Run2 @ S2 )
                = ( some_P4227418909054022151_d_nat @ ( produc5865182755557626534_d_nat @ S8 @ V ) ) )
              & ( reache5148256237901502841_d_nat @ Run2 @ S8 @ Vs3 @ S9 ) ) ) ) ) ).

% reaches_on.simps
thf(fact_995_reaches__on_Osimps,axiom,
    ( reaches_on_a_b
    = ( ^ [Run2: a > option5281194463001729774od_a_b,A12: a,A23: list_b,A33: a] :
          ( ? [S2: a] :
              ( ( A12 = S2 )
              & ( A23 = nil_b )
              & ( A33 = S2 ) )
          | ? [S2: a,S8: a,V: b,Vs3: list_b,S9: a] :
              ( ( A12 = S2 )
              & ( A23
                = ( cons_b @ V @ Vs3 ) )
              & ( A33 = S9 )
              & ( ( Run2 @ S2 )
                = ( some_P3592067299498605709od_a_b @ ( product_Pair_a_b @ S8 @ V ) ) )
              & ( reaches_on_a_b @ Run2 @ S8 @ Vs3 @ S9 ) ) ) ) ) ).

% reaches_on.simps
thf(fact_996_reaches__on_Osimps,axiom,
    ( reache6877781913633749273_a_b_a
    = ( ^ [Run2: produc1990351791824763820_d_nat > option2221477324614387130_a_b_a,A12: produc1990351791824763820_d_nat,A23: list_m9210641325446214647_a_b_a,A33: produc1990351791824763820_d_nat] :
          ( ? [S2: produc1990351791824763820_d_nat] :
              ( ( A12 = S2 )
              & ( A23 = nil_ma546712388551201761_a_b_a )
              & ( A33 = S2 ) )
          | ? [S2: produc1990351791824763820_d_nat,S8: produc1990351791824763820_d_nat,V: mappin2365780136655028071_a_b_a,Vs3: list_m9210641325446214647_a_b_a,S9: produc1990351791824763820_d_nat] :
              ( ( A12 = S2 )
              & ( A23
                = ( cons_m8574943908289518641_a_b_a @ V @ Vs3 ) )
              & ( A33 = S9 )
              & ( ( Run2 @ S2 )
                = ( some_P6648320236558503769_a_b_a @ ( produc1681437058057156268_a_b_a @ S8 @ V ) ) )
              & ( reache6877781913633749273_a_b_a @ Run2 @ S8 @ Vs3 @ S9 ) ) ) ) ) ).

% reaches_on.simps
thf(fact_997_reaches__on_Osimps,axiom,
    ( reache4265786078874959145_a_b_a
    = ( ^ [Run2: list_P6684558232516339260_d_nat > option4722335503605243466_a_b_a,A12: list_P6684558232516339260_d_nat,A23: list_m9210641325446214647_a_b_a,A33: list_P6684558232516339260_d_nat] :
          ( ? [S2: list_P6684558232516339260_d_nat] :
              ( ( A12 = S2 )
              & ( A23 = nil_ma546712388551201761_a_b_a )
              & ( A33 = S2 ) )
          | ? [S2: list_P6684558232516339260_d_nat,S8: list_P6684558232516339260_d_nat,V: mappin2365780136655028071_a_b_a,Vs3: list_m9210641325446214647_a_b_a,S9: list_P6684558232516339260_d_nat] :
              ( ( A12 = S2 )
              & ( A23
                = ( cons_m8574943908289518641_a_b_a @ V @ Vs3 ) )
              & ( A33 = S9 )
              & ( ( Run2 @ S2 )
                = ( some_P4038468990576657129_a_b_a @ ( produc2616710729536263484_a_b_a @ S8 @ V ) ) )
              & ( reache4265786078874959145_a_b_a @ Run2 @ S8 @ Vs3 @ S9 ) ) ) ) ) ).

% reaches_on.simps
thf(fact_998_reaches__on_Osimps,axiom,
    ( reache8162303588387466849_a_b_a
    = ( ^ [Run2: a > option3723494939309271012_a_b_a,A12: a,A23: list_m9210641325446214647_a_b_a,A33: a] :
          ( ? [S2: a] :
              ( ( A12 = S2 )
              & ( A23 = nil_ma546712388551201761_a_b_a )
              & ( A33 = S2 ) )
          | ? [S2: a,S8: a,V: mappin2365780136655028071_a_b_a,Vs3: list_m9210641325446214647_a_b_a,S9: a] :
              ( ( A12 = S2 )
              & ( A23
                = ( cons_m8574943908289518641_a_b_a @ V @ Vs3 ) )
              & ( A33 = S9 )
              & ( ( Run2 @ S2 )
                = ( some_P2288122801350569071_a_b_a @ ( produc1583556749610718734_a_b_a @ S8 @ V ) ) )
              & ( reache8162303588387466849_a_b_a @ Run2 @ S8 @ Vs3 @ S9 ) ) ) ) ) ).

% reaches_on.simps
thf(fact_999_reaches__on__app,axiom,
    ! [Run: a > option3982279322286820476_d_nat,S3: a,Vs: list_P7053655179267667343_d_nat,S4: a,S6: a,V2: produc5456268773499131775_d_nat] :
      ( ( reache5148256237901502841_d_nat @ Run @ S3 @ Vs @ S4 )
     => ( ( ( Run @ S4 )
          = ( some_P4227418909054022151_d_nat @ ( produc5865182755557626534_d_nat @ S6 @ V2 ) ) )
       => ( reache5148256237901502841_d_nat @ Run @ S3 @ ( append5329052816365141668_d_nat @ Vs @ ( cons_P8522577077107056457_d_nat @ V2 @ nil_Pr3001256489040166649_d_nat ) ) @ S6 ) ) ) ).

% reaches_on_app
thf(fact_1000_reaches__on__app,axiom,
    ! [Run: a > option5281194463001729774od_a_b,S3: a,Vs: list_b,S4: a,S6: a,V2: b] :
      ( ( reaches_on_a_b @ Run @ S3 @ Vs @ S4 )
     => ( ( ( Run @ S4 )
          = ( some_P3592067299498605709od_a_b @ ( product_Pair_a_b @ S6 @ V2 ) ) )
       => ( reaches_on_a_b @ Run @ S3 @ ( append_b @ Vs @ ( cons_b @ V2 @ nil_b ) ) @ S6 ) ) ) ).

% reaches_on_app
thf(fact_1001_reaches__on__app,axiom,
    ! [Run: produc1990351791824763820_d_nat > option2221477324614387130_a_b_a,S3: produc1990351791824763820_d_nat,Vs: list_m9210641325446214647_a_b_a,S4: produc1990351791824763820_d_nat,S6: produc1990351791824763820_d_nat,V2: mappin2365780136655028071_a_b_a] :
      ( ( reache6877781913633749273_a_b_a @ Run @ S3 @ Vs @ S4 )
     => ( ( ( Run @ S4 )
          = ( some_P6648320236558503769_a_b_a @ ( produc1681437058057156268_a_b_a @ S6 @ V2 ) ) )
       => ( reache6877781913633749273_a_b_a @ Run @ S3 @ ( append7379309756631650572_a_b_a @ Vs @ ( cons_m8574943908289518641_a_b_a @ V2 @ nil_ma546712388551201761_a_b_a ) ) @ S6 ) ) ) ).

% reaches_on_app
thf(fact_1002_reaches__on__app,axiom,
    ! [Run: list_P6684558232516339260_d_nat > option4722335503605243466_a_b_a,S3: list_P6684558232516339260_d_nat,Vs: list_m9210641325446214647_a_b_a,S4: list_P6684558232516339260_d_nat,S6: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a] :
      ( ( reache4265786078874959145_a_b_a @ Run @ S3 @ Vs @ S4 )
     => ( ( ( Run @ S4 )
          = ( some_P4038468990576657129_a_b_a @ ( produc2616710729536263484_a_b_a @ S6 @ V2 ) ) )
       => ( reache4265786078874959145_a_b_a @ Run @ S3 @ ( append7379309756631650572_a_b_a @ Vs @ ( cons_m8574943908289518641_a_b_a @ V2 @ nil_ma546712388551201761_a_b_a ) ) @ S6 ) ) ) ).

% reaches_on_app
thf(fact_1003_reaches__on__app,axiom,
    ! [Run: a > option3723494939309271012_a_b_a,S3: a,Vs: list_m9210641325446214647_a_b_a,S4: a,S6: a,V2: mappin2365780136655028071_a_b_a] :
      ( ( reache8162303588387466849_a_b_a @ Run @ S3 @ Vs @ S4 )
     => ( ( ( Run @ S4 )
          = ( some_P2288122801350569071_a_b_a @ ( produc1583556749610718734_a_b_a @ S6 @ V2 ) ) )
       => ( reache8162303588387466849_a_b_a @ Run @ S3 @ ( append7379309756631650572_a_b_a @ Vs @ ( cons_m8574943908289518641_a_b_a @ V2 @ nil_ma546712388551201761_a_b_a ) ) @ S6 ) ) ) ).

% reaches_on_app
thf(fact_1004_reaches__on__rev__induct,axiom,
    ! [Run: a > option3982279322286820476_d_nat,S3: a,Vs: list_P7053655179267667343_d_nat,S4: a,P: a > list_P7053655179267667343_d_nat > a > $o] :
      ( ( reache5148256237901502841_d_nat @ Run @ S3 @ Vs @ S4 )
     => ( ! [S10: a] : ( P @ S10 @ nil_Pr3001256489040166649_d_nat @ S10 )
       => ( ! [S10: a,S7: a,V4: produc5456268773499131775_d_nat,Vs2: list_P7053655179267667343_d_nat,S5: a] :
              ( ( reache5148256237901502841_d_nat @ Run @ S10 @ Vs2 @ S7 )
             => ( ( P @ S10 @ Vs2 @ S7 )
               => ( ( ( Run @ S7 )
                    = ( some_P4227418909054022151_d_nat @ ( produc5865182755557626534_d_nat @ S5 @ V4 ) ) )
                 => ( P @ S10 @ ( append5329052816365141668_d_nat @ Vs2 @ ( cons_P8522577077107056457_d_nat @ V4 @ nil_Pr3001256489040166649_d_nat ) ) @ S5 ) ) ) )
         => ( P @ S3 @ Vs @ S4 ) ) ) ) ).

% reaches_on_rev_induct
thf(fact_1005_reaches__on__rev__induct,axiom,
    ! [Run: a > option5281194463001729774od_a_b,S3: a,Vs: list_b,S4: a,P: a > list_b > a > $o] :
      ( ( reaches_on_a_b @ Run @ S3 @ Vs @ S4 )
     => ( ! [S10: a] : ( P @ S10 @ nil_b @ S10 )
       => ( ! [S10: a,S7: a,V4: b,Vs2: list_b,S5: a] :
              ( ( reaches_on_a_b @ Run @ S10 @ Vs2 @ S7 )
             => ( ( P @ S10 @ Vs2 @ S7 )
               => ( ( ( Run @ S7 )
                    = ( some_P3592067299498605709od_a_b @ ( product_Pair_a_b @ S5 @ V4 ) ) )
                 => ( P @ S10 @ ( append_b @ Vs2 @ ( cons_b @ V4 @ nil_b ) ) @ S5 ) ) ) )
         => ( P @ S3 @ Vs @ S4 ) ) ) ) ).

% reaches_on_rev_induct
thf(fact_1006_reaches__on__rev__induct,axiom,
    ! [Run: produc1990351791824763820_d_nat > option2221477324614387130_a_b_a,S3: produc1990351791824763820_d_nat,Vs: list_m9210641325446214647_a_b_a,S4: produc1990351791824763820_d_nat,P: produc1990351791824763820_d_nat > list_m9210641325446214647_a_b_a > produc1990351791824763820_d_nat > $o] :
      ( ( reache6877781913633749273_a_b_a @ Run @ S3 @ Vs @ S4 )
     => ( ! [S10: produc1990351791824763820_d_nat] : ( P @ S10 @ nil_ma546712388551201761_a_b_a @ S10 )
       => ( ! [S10: produc1990351791824763820_d_nat,S7: produc1990351791824763820_d_nat,V4: mappin2365780136655028071_a_b_a,Vs2: list_m9210641325446214647_a_b_a,S5: produc1990351791824763820_d_nat] :
              ( ( reache6877781913633749273_a_b_a @ Run @ S10 @ Vs2 @ S7 )
             => ( ( P @ S10 @ Vs2 @ S7 )
               => ( ( ( Run @ S7 )
                    = ( some_P6648320236558503769_a_b_a @ ( produc1681437058057156268_a_b_a @ S5 @ V4 ) ) )
                 => ( P @ S10 @ ( append7379309756631650572_a_b_a @ Vs2 @ ( cons_m8574943908289518641_a_b_a @ V4 @ nil_ma546712388551201761_a_b_a ) ) @ S5 ) ) ) )
         => ( P @ S3 @ Vs @ S4 ) ) ) ) ).

% reaches_on_rev_induct
thf(fact_1007_reaches__on__rev__induct,axiom,
    ! [Run: list_P6684558232516339260_d_nat > option4722335503605243466_a_b_a,S3: list_P6684558232516339260_d_nat,Vs: list_m9210641325446214647_a_b_a,S4: list_P6684558232516339260_d_nat,P: list_P6684558232516339260_d_nat > list_m9210641325446214647_a_b_a > list_P6684558232516339260_d_nat > $o] :
      ( ( reache4265786078874959145_a_b_a @ Run @ S3 @ Vs @ S4 )
     => ( ! [S10: list_P6684558232516339260_d_nat] : ( P @ S10 @ nil_ma546712388551201761_a_b_a @ S10 )
       => ( ! [S10: list_P6684558232516339260_d_nat,S7: list_P6684558232516339260_d_nat,V4: mappin2365780136655028071_a_b_a,Vs2: list_m9210641325446214647_a_b_a,S5: list_P6684558232516339260_d_nat] :
              ( ( reache4265786078874959145_a_b_a @ Run @ S10 @ Vs2 @ S7 )
             => ( ( P @ S10 @ Vs2 @ S7 )
               => ( ( ( Run @ S7 )
                    = ( some_P4038468990576657129_a_b_a @ ( produc2616710729536263484_a_b_a @ S5 @ V4 ) ) )
                 => ( P @ S10 @ ( append7379309756631650572_a_b_a @ Vs2 @ ( cons_m8574943908289518641_a_b_a @ V4 @ nil_ma546712388551201761_a_b_a ) ) @ S5 ) ) ) )
         => ( P @ S3 @ Vs @ S4 ) ) ) ) ).

% reaches_on_rev_induct
thf(fact_1008_reaches__on__rev__induct,axiom,
    ! [Run: a > option3723494939309271012_a_b_a,S3: a,Vs: list_m9210641325446214647_a_b_a,S4: a,P: a > list_m9210641325446214647_a_b_a > a > $o] :
      ( ( reache8162303588387466849_a_b_a @ Run @ S3 @ Vs @ S4 )
     => ( ! [S10: a] : ( P @ S10 @ nil_ma546712388551201761_a_b_a @ S10 )
       => ( ! [S10: a,S7: a,V4: mappin2365780136655028071_a_b_a,Vs2: list_m9210641325446214647_a_b_a,S5: a] :
              ( ( reache8162303588387466849_a_b_a @ Run @ S10 @ Vs2 @ S7 )
             => ( ( P @ S10 @ Vs2 @ S7 )
               => ( ( ( Run @ S7 )
                    = ( some_P2288122801350569071_a_b_a @ ( produc1583556749610718734_a_b_a @ S5 @ V4 ) ) )
                 => ( P @ S10 @ ( append7379309756631650572_a_b_a @ Vs2 @ ( cons_m8574943908289518641_a_b_a @ V4 @ nil_ma546712388551201761_a_b_a ) ) @ S5 ) ) ) )
         => ( P @ S3 @ Vs @ S4 ) ) ) ) ).

% reaches_on_rev_induct
thf(fact_1009_reaches__on__split__last,axiom,
    ! [Run: a > option3982279322286820476_d_nat,S3: a,Xs2: list_P7053655179267667343_d_nat,X: produc5456268773499131775_d_nat,S6: a] :
      ( ( reache5148256237901502841_d_nat @ Run @ S3 @ ( append5329052816365141668_d_nat @ Xs2 @ ( cons_P8522577077107056457_d_nat @ X @ nil_Pr3001256489040166649_d_nat ) ) @ S6 )
     => ? [S7: a] :
          ( ( reache5148256237901502841_d_nat @ Run @ S3 @ Xs2 @ S7 )
          & ( ( Run @ S7 )
            = ( some_P4227418909054022151_d_nat @ ( produc5865182755557626534_d_nat @ S6 @ X ) ) ) ) ) ).

% reaches_on_split_last
thf(fact_1010_reaches__on__split__last,axiom,
    ! [Run: a > option5281194463001729774od_a_b,S3: a,Xs2: list_b,X: b,S6: a] :
      ( ( reaches_on_a_b @ Run @ S3 @ ( append_b @ Xs2 @ ( cons_b @ X @ nil_b ) ) @ S6 )
     => ? [S7: a] :
          ( ( reaches_on_a_b @ Run @ S3 @ Xs2 @ S7 )
          & ( ( Run @ S7 )
            = ( some_P3592067299498605709od_a_b @ ( product_Pair_a_b @ S6 @ X ) ) ) ) ) ).

% reaches_on_split_last
thf(fact_1011_reaches__on__split__last,axiom,
    ! [Run: produc1990351791824763820_d_nat > option2221477324614387130_a_b_a,S3: produc1990351791824763820_d_nat,Xs2: list_m9210641325446214647_a_b_a,X: mappin2365780136655028071_a_b_a,S6: produc1990351791824763820_d_nat] :
      ( ( reache6877781913633749273_a_b_a @ Run @ S3 @ ( append7379309756631650572_a_b_a @ Xs2 @ ( cons_m8574943908289518641_a_b_a @ X @ nil_ma546712388551201761_a_b_a ) ) @ S6 )
     => ? [S7: produc1990351791824763820_d_nat] :
          ( ( reache6877781913633749273_a_b_a @ Run @ S3 @ Xs2 @ S7 )
          & ( ( Run @ S7 )
            = ( some_P6648320236558503769_a_b_a @ ( produc1681437058057156268_a_b_a @ S6 @ X ) ) ) ) ) ).

% reaches_on_split_last
thf(fact_1012_reaches__on__split__last,axiom,
    ! [Run: list_P6684558232516339260_d_nat > option4722335503605243466_a_b_a,S3: list_P6684558232516339260_d_nat,Xs2: list_m9210641325446214647_a_b_a,X: mappin2365780136655028071_a_b_a,S6: list_P6684558232516339260_d_nat] :
      ( ( reache4265786078874959145_a_b_a @ Run @ S3 @ ( append7379309756631650572_a_b_a @ Xs2 @ ( cons_m8574943908289518641_a_b_a @ X @ nil_ma546712388551201761_a_b_a ) ) @ S6 )
     => ? [S7: list_P6684558232516339260_d_nat] :
          ( ( reache4265786078874959145_a_b_a @ Run @ S3 @ Xs2 @ S7 )
          & ( ( Run @ S7 )
            = ( some_P4038468990576657129_a_b_a @ ( produc2616710729536263484_a_b_a @ S6 @ X ) ) ) ) ) ).

% reaches_on_split_last
thf(fact_1013_reaches__on__split__last,axiom,
    ! [Run: a > option3723494939309271012_a_b_a,S3: a,Xs2: list_m9210641325446214647_a_b_a,X: mappin2365780136655028071_a_b_a,S6: a] :
      ( ( reache8162303588387466849_a_b_a @ Run @ S3 @ ( append7379309756631650572_a_b_a @ Xs2 @ ( cons_m8574943908289518641_a_b_a @ X @ nil_ma546712388551201761_a_b_a ) ) @ S6 )
     => ? [S7: a] :
          ( ( reache8162303588387466849_a_b_a @ Run @ S3 @ Xs2 @ S7 )
          & ( ( Run @ S7 )
            = ( some_P2288122801350569071_a_b_a @ ( produc1583556749610718734_a_b_a @ S6 @ X ) ) ) ) ) ).

% reaches_on_split_last
thf(fact_1014_entries__update,axiom,
    ! [K2: a,V2: produc5456268773499131775_d_nat,M: mappin6535725686296717250_d_nat] :
      ( ( entrie8866367667353307151_d_nat @ ( update6116250764560905298_d_nat @ K2 @ V2 @ M ) )
      = ( insert934994597402771932_d_nat @ ( produc5865182755557626534_d_nat @ K2 @ V2 ) @ ( entrie8866367667353307151_d_nat @ ( delete6742496776821646864_d_nat @ K2 @ M ) ) ) ) ).

% entries_update
thf(fact_1015_entries__update,axiom,
    ! [K2: a,V2: b,M: mapping_a_b] :
      ( ( entries_a_b @ ( update_a_b @ K2 @ V2 @ M ) )
      = ( insert4534936386344385144od_a_b @ ( product_Pair_a_b @ K2 @ V2 ) @ ( entries_a_b @ ( delete_a_b @ K2 @ M ) ) ) ) ).

% entries_update
thf(fact_1016_entries__update,axiom,
    ! [K2: produc1990351791824763820_d_nat,V2: mappin2365780136655028071_a_b_a,M: mappin1208405706615879902_a_b_a] :
      ( ( entrie369595780118757635_a_b_a @ ( update8633133774124818944_a_b_a @ K2 @ V2 @ M ) )
      = ( insert1602870736147291204_a_b_a @ ( produc1681437058057156268_a_b_a @ K2 @ V2 ) @ ( entrie369595780118757635_a_b_a @ ( delete5632990397533007042_a_b_a @ K2 @ M ) ) ) ) ).

% entries_update
thf(fact_1017_entries__update,axiom,
    ! [K2: list_P6684558232516339260_d_nat,V2: mappin2365780136655028071_a_b_a,M: mappin187059948859696622_a_b_a] :
      ( ( entrie4359402741916977171_a_b_a @ ( update5108779570945132176_a_b_a @ K2 @ V2 @ M ) )
      = ( insert6197680461883956820_a_b_a @ ( produc2616710729536263484_a_b_a @ K2 @ V2 ) @ ( entrie4359402741916977171_a_b_a @ ( delete6178897237033142354_a_b_a @ K2 @ M ) ) ) ) ).

% entries_update
thf(fact_1018_entries__update,axiom,
    ! [K2: a,V2: mappin2365780136655028071_a_b_a,M: mappin4519906018992269226_a_b_a] :
      ( ( entrie7794251174888650743_a_b_a @ ( update5632971445410033082_a_b_a @ K2 @ V2 @ M ) )
      = ( insert1871085374464671428_a_b_a @ ( produc1583556749610718734_a_b_a @ K2 @ V2 ) @ ( entrie7794251174888650743_a_b_a @ ( delete9073802908114374264_a_b_a @ K2 @ M ) ) ) ) ).

% entries_update
thf(fact_1019_All__mapping__alist,axiom,
    ! [Xs2: list_P6684558232516339260_d_nat,P: a > produc5456268773499131775_d_nat > $o] :
      ( ! [K5: a,V4: produc5456268773499131775_d_nat] :
          ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ K5 @ V4 ) @ ( set_Pr1108108036290734423_d_nat @ Xs2 ) )
         => ( P @ K5 @ V4 ) )
     => ( all_ma1625059968680912654_d_nat @ ( of_ali1702328068607691863_d_nat @ Xs2 ) @ P ) ) ).

% All_mapping_alist
thf(fact_1020_All__mapping__alist,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,P: a > b > $o] :
      ( ! [K5: a,V4: b] :
          ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ K5 @ V4 ) @ ( set_Product_prod_a_b2 @ Xs2 ) )
         => ( P @ K5 @ V4 ) )
     => ( all_mapping_a_b @ ( of_alist_a_b @ Xs2 ) @ P ) ) ).

% All_mapping_alist
thf(fact_1021_All__mapping__alist,axiom,
    ! [Xs2: list_P4793483941206178810_a_b_a,P: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > $o] :
      ( ! [K5: produc1990351791824763820_d_nat,V4: mappin2365780136655028071_a_b_a] :
          ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ K5 @ V4 ) @ ( set_Pr3502879326370882569_a_b_a @ Xs2 ) )
         => ( P @ K5 @ V4 ) )
     => ( all_ma8096219400971294788_a_b_a @ ( of_ali2289480013228440251_a_b_a @ Xs2 ) @ P ) ) ).

% All_mapping_alist
thf(fact_1022_All__mapping__alist,axiom,
    ! [Xs2: list_P7611574810425128074_a_b_a,P: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > $o] :
      ( ! [K5: list_P6684558232516339260_d_nat,V4: mappin2365780136655028071_a_b_a] :
          ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ K5 @ V4 ) @ ( set_Pr4840226057646735257_a_b_a @ Xs2 ) )
         => ( P @ K5 @ V4 ) )
     => ( all_ma3172627230583229140_a_b_a @ ( of_ali7156197523171788235_a_b_a @ Xs2 ) @ P ) ) ).

% All_mapping_alist
thf(fact_1023_All__mapping__alist,axiom,
    ! [Xs2: list_P2992919015626629540_a_b_a,P: a > mappin2365780136655028071_a_b_a > $o] :
      ( ! [K5: a,V4: mappin2365780136655028071_a_b_a] :
          ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ K5 @ V4 ) @ ( set_Pr7630203459061173183_a_b_a @ Xs2 ) )
         => ( P @ K5 @ V4 ) )
     => ( all_ma2913212136517639798_a_b_a @ ( of_ali8740168035790364735_a_b_a @ Xs2 ) @ P ) ) ).

% All_mapping_alist
thf(fact_1024_Range__insert,axiom,
    ! [A: a,B: produc5456268773499131775_d_nat,R3: set_Pr8583615357880026082_d_nat] :
      ( ( range_1428461333860462920_d_nat @ ( insert934994597402771932_d_nat @ ( produc5865182755557626534_d_nat @ A @ B ) @ R3 ) )
      = ( insert8628257856670502319_d_nat @ B @ ( range_1428461333860462920_d_nat @ R3 ) ) ) ).

% Range_insert
thf(fact_1025_Range__insert,axiom,
    ! [A: a,B: b,R3: set_Product_prod_a_b] :
      ( ( range_a_b @ ( insert4534936386344385144od_a_b @ ( product_Pair_a_b @ A @ B ) @ R3 ) )
      = ( insert_b2 @ B @ ( range_a_b @ R3 ) ) ) ).

% Range_insert
thf(fact_1026_Range__insert,axiom,
    ! [A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a,R3: set_Pr8226792695654363348_a_b_a] :
      ( ( range_8045137746665472906_a_b_a @ ( insert1602870736147291204_a_b_a @ ( produc1681437058057156268_a_b_a @ A @ B ) @ R3 ) )
      = ( insert2272198572858264983_a_b_a @ B @ ( range_8045137746665472906_a_b_a @ R3 ) ) ) ).

% Range_insert
thf(fact_1027_Range__insert,axiom,
    ! [A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a,R3: set_Pr2043227878544749668_a_b_a] :
      ( ( range_5113499372716530458_a_b_a @ ( insert6197680461883956820_a_b_a @ ( produc2616710729536263484_a_b_a @ A @ B ) @ R3 ) )
      = ( insert2272198572858264983_a_b_a @ B @ ( range_5113499372716530458_a_b_a @ R3 ) ) ) ).

% Range_insert
thf(fact_1028_Range__insert,axiom,
    ! [A: a,B: mappin2365780136655028071_a_b_a,R3: set_Pr2084289749507318346_a_b_a] :
      ( ( range_5462226226884687280_a_b_a @ ( insert1871085374464671428_a_b_a @ ( produc1583556749610718734_a_b_a @ A @ B ) @ R3 ) )
      = ( insert2272198572858264983_a_b_a @ B @ ( range_5462226226884687280_a_b_a @ R3 ) ) ) ).

% Range_insert
thf(fact_1029_Range__iff,axiom,
    ! [A: produc5456268773499131775_d_nat,R3: set_Pr8583615357880026082_d_nat] :
      ( ( member8156885303890938774_d_nat @ A @ ( range_1428461333860462920_d_nat @ R3 ) )
      = ( ? [Y4: a] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ Y4 @ A ) @ R3 ) ) ) ).

% Range_iff
thf(fact_1030_Range__iff,axiom,
    ! [A: b,R3: set_Product_prod_a_b] :
      ( ( member_b @ A @ ( range_a_b @ R3 ) )
      = ( ? [Y4: a] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ Y4 @ A ) @ R3 ) ) ) ).

% Range_iff
thf(fact_1031_Range__iff,axiom,
    ! [A: mappin2365780136655028071_a_b_a,R3: set_Pr8226792695654363348_a_b_a] :
      ( ( member2890181943093937918_a_b_a @ A @ ( range_8045137746665472906_a_b_a @ R3 ) )
      = ( ? [Y4: produc1990351791824763820_d_nat] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ Y4 @ A ) @ R3 ) ) ) ).

% Range_iff
thf(fact_1032_Range__iff,axiom,
    ! [A: mappin2365780136655028071_a_b_a,R3: set_Pr2043227878544749668_a_b_a] :
      ( ( member2890181943093937918_a_b_a @ A @ ( range_5113499372716530458_a_b_a @ R3 ) )
      = ( ? [Y4: list_P6684558232516339260_d_nat] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ Y4 @ A ) @ R3 ) ) ) ).

% Range_iff
thf(fact_1033_Range__iff,axiom,
    ! [A: mappin2365780136655028071_a_b_a,R3: set_Pr2084289749507318346_a_b_a] :
      ( ( member2890181943093937918_a_b_a @ A @ ( range_5462226226884687280_a_b_a @ R3 ) )
      = ( ? [Y4: a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ Y4 @ A ) @ R3 ) ) ) ).

% Range_iff
thf(fact_1034_RangeE,axiom,
    ! [B: produc5456268773499131775_d_nat,R3: set_Pr8583615357880026082_d_nat] :
      ( ( member8156885303890938774_d_nat @ B @ ( range_1428461333860462920_d_nat @ R3 ) )
     => ~ ! [A3: a] :
            ~ ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ A3 @ B ) @ R3 ) ) ).

% RangeE
thf(fact_1035_RangeE,axiom,
    ! [B: b,R3: set_Product_prod_a_b] :
      ( ( member_b @ B @ ( range_a_b @ R3 ) )
     => ~ ! [A3: a] :
            ~ ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A3 @ B ) @ R3 ) ) ).

% RangeE
thf(fact_1036_RangeE,axiom,
    ! [B: mappin2365780136655028071_a_b_a,R3: set_Pr8226792695654363348_a_b_a] :
      ( ( member2890181943093937918_a_b_a @ B @ ( range_8045137746665472906_a_b_a @ R3 ) )
     => ~ ! [A3: produc1990351791824763820_d_nat] :
            ~ ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ A3 @ B ) @ R3 ) ) ).

% RangeE
thf(fact_1037_RangeE,axiom,
    ! [B: mappin2365780136655028071_a_b_a,R3: set_Pr2043227878544749668_a_b_a] :
      ( ( member2890181943093937918_a_b_a @ B @ ( range_5113499372716530458_a_b_a @ R3 ) )
     => ~ ! [A3: list_P6684558232516339260_d_nat] :
            ~ ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ A3 @ B ) @ R3 ) ) ).

% RangeE
thf(fact_1038_RangeE,axiom,
    ! [B: mappin2365780136655028071_a_b_a,R3: set_Pr2084289749507318346_a_b_a] :
      ( ( member2890181943093937918_a_b_a @ B @ ( range_5462226226884687280_a_b_a @ R3 ) )
     => ~ ! [A3: a] :
            ~ ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ A3 @ B ) @ R3 ) ) ).

% RangeE
thf(fact_1039_Range_Ointros,axiom,
    ! [A: a,B: produc5456268773499131775_d_nat,R3: set_Pr8583615357880026082_d_nat] :
      ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ A @ B ) @ R3 )
     => ( member8156885303890938774_d_nat @ B @ ( range_1428461333860462920_d_nat @ R3 ) ) ) ).

% Range.intros
thf(fact_1040_Range_Ointros,axiom,
    ! [A: a,B: b,R3: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B ) @ R3 )
     => ( member_b @ B @ ( range_a_b @ R3 ) ) ) ).

% Range.intros
thf(fact_1041_Range_Ointros,axiom,
    ! [A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a,R3: set_Pr8226792695654363348_a_b_a] :
      ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ A @ B ) @ R3 )
     => ( member2890181943093937918_a_b_a @ B @ ( range_8045137746665472906_a_b_a @ R3 ) ) ) ).

% Range.intros
thf(fact_1042_Range_Ointros,axiom,
    ! [A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a,R3: set_Pr2043227878544749668_a_b_a] :
      ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ A @ B ) @ R3 )
     => ( member2890181943093937918_a_b_a @ B @ ( range_5113499372716530458_a_b_a @ R3 ) ) ) ).

% Range.intros
thf(fact_1043_Range_Ointros,axiom,
    ! [A: a,B: mappin2365780136655028071_a_b_a,R3: set_Pr2084289749507318346_a_b_a] :
      ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ A @ B ) @ R3 )
     => ( member2890181943093937918_a_b_a @ B @ ( range_5462226226884687280_a_b_a @ R3 ) ) ) ).

% Range.intros
thf(fact_1044_Range_Osimps,axiom,
    ! [A: produc5456268773499131775_d_nat,R3: set_Pr8583615357880026082_d_nat] :
      ( ( member8156885303890938774_d_nat @ A @ ( range_1428461333860462920_d_nat @ R3 ) )
      = ( ? [A5: a,B4: produc5456268773499131775_d_nat] :
            ( ( A = B4 )
            & ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ A5 @ B4 ) @ R3 ) ) ) ) ).

% Range.simps
thf(fact_1045_Range_Osimps,axiom,
    ! [A: b,R3: set_Product_prod_a_b] :
      ( ( member_b @ A @ ( range_a_b @ R3 ) )
      = ( ? [A5: a,B4: b] :
            ( ( A = B4 )
            & ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A5 @ B4 ) @ R3 ) ) ) ) ).

% Range.simps
thf(fact_1046_Range_Osimps,axiom,
    ! [A: mappin2365780136655028071_a_b_a,R3: set_Pr8226792695654363348_a_b_a] :
      ( ( member2890181943093937918_a_b_a @ A @ ( range_8045137746665472906_a_b_a @ R3 ) )
      = ( ? [A5: produc1990351791824763820_d_nat,B4: mappin2365780136655028071_a_b_a] :
            ( ( A = B4 )
            & ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ A5 @ B4 ) @ R3 ) ) ) ) ).

% Range.simps
thf(fact_1047_Range_Osimps,axiom,
    ! [A: mappin2365780136655028071_a_b_a,R3: set_Pr2043227878544749668_a_b_a] :
      ( ( member2890181943093937918_a_b_a @ A @ ( range_5113499372716530458_a_b_a @ R3 ) )
      = ( ? [A5: list_P6684558232516339260_d_nat,B4: mappin2365780136655028071_a_b_a] :
            ( ( A = B4 )
            & ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ A5 @ B4 ) @ R3 ) ) ) ) ).

% Range.simps
thf(fact_1048_Range_Osimps,axiom,
    ! [A: mappin2365780136655028071_a_b_a,R3: set_Pr2084289749507318346_a_b_a] :
      ( ( member2890181943093937918_a_b_a @ A @ ( range_5462226226884687280_a_b_a @ R3 ) )
      = ( ? [A5: a,B4: mappin2365780136655028071_a_b_a] :
            ( ( A = B4 )
            & ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ A5 @ B4 ) @ R3 ) ) ) ) ).

% Range.simps
thf(fact_1049_Range_Ocases,axiom,
    ! [A: produc5456268773499131775_d_nat,R3: set_Pr8583615357880026082_d_nat] :
      ( ( member8156885303890938774_d_nat @ A @ ( range_1428461333860462920_d_nat @ R3 ) )
     => ~ ! [A3: a] :
            ~ ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ A3 @ A ) @ R3 ) ) ).

% Range.cases
thf(fact_1050_Range_Ocases,axiom,
    ! [A: b,R3: set_Product_prod_a_b] :
      ( ( member_b @ A @ ( range_a_b @ R3 ) )
     => ~ ! [A3: a] :
            ~ ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A3 @ A ) @ R3 ) ) ).

% Range.cases
thf(fact_1051_Range_Ocases,axiom,
    ! [A: mappin2365780136655028071_a_b_a,R3: set_Pr8226792695654363348_a_b_a] :
      ( ( member2890181943093937918_a_b_a @ A @ ( range_8045137746665472906_a_b_a @ R3 ) )
     => ~ ! [A3: produc1990351791824763820_d_nat] :
            ~ ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ A3 @ A ) @ R3 ) ) ).

% Range.cases
thf(fact_1052_Range_Ocases,axiom,
    ! [A: mappin2365780136655028071_a_b_a,R3: set_Pr2043227878544749668_a_b_a] :
      ( ( member2890181943093937918_a_b_a @ A @ ( range_5113499372716530458_a_b_a @ R3 ) )
     => ~ ! [A3: list_P6684558232516339260_d_nat] :
            ~ ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ A3 @ A ) @ R3 ) ) ).

% Range.cases
thf(fact_1053_Range_Ocases,axiom,
    ! [A: mappin2365780136655028071_a_b_a,R3: set_Pr2084289749507318346_a_b_a] :
      ( ( member2890181943093937918_a_b_a @ A @ ( range_5462226226884687280_a_b_a @ R3 ) )
     => ~ ! [A3: a] :
            ~ ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ A3 @ A ) @ R3 ) ) ).

% Range.cases
thf(fact_1054_prod__set__simps_I1_J,axiom,
    ! [X: a,Y5: produc5456268773499131775_d_nat] :
      ( ( basic_456393284961243574_d_nat @ ( produc5865182755557626534_d_nat @ X @ Y5 ) )
      = ( insert_a2 @ X @ bot_bot_set_a ) ) ).

% prod_set_simps(1)
thf(fact_1055_prod__set__simps_I1_J,axiom,
    ! [X: a,Y5: b] :
      ( ( basic_fsts_a_b @ ( product_Pair_a_b @ X @ Y5 ) )
      = ( insert_a2 @ X @ bot_bot_set_a ) ) ).

% prod_set_simps(1)
thf(fact_1056_prod__set__simps_I1_J,axiom,
    ! [X: produc1990351791824763820_d_nat,Y5: mappin2365780136655028071_a_b_a] :
      ( ( basic_4692323668668502940_a_b_a @ ( produc1681437058057156268_a_b_a @ X @ Y5 ) )
      = ( insert934994597402771932_d_nat @ X @ bot_bo534106183387690102_d_nat ) ) ).

% prod_set_simps(1)
thf(fact_1057_prod__set__simps_I1_J,axiom,
    ! [X: list_P6684558232516339260_d_nat,Y5: mappin2365780136655028071_a_b_a] :
      ( ( basic_8180497794411477548_a_b_a @ ( produc2616710729536263484_a_b_a @ X @ Y5 ) )
      = ( insert5227142572749542508_d_nat @ X @ bot_bo7515912283962571654_d_nat ) ) ).

% prod_set_simps(1)
thf(fact_1058_prod__set__simps_I1_J,axiom,
    ! [X: a,Y5: mappin2365780136655028071_a_b_a] :
      ( ( basic_6361707675047025950_a_b_a @ ( produc1583556749610718734_a_b_a @ X @ Y5 ) )
      = ( insert_a2 @ X @ bot_bot_set_a ) ) ).

% prod_set_simps(1)
thf(fact_1059_Range__def,axiom,
    ( range_1428461333860462920_d_nat
    = ( ^ [R2: set_Pr8583615357880026082_d_nat] :
          ( collec4764653592595650004_d_nat
          @ ( rangep7060683137087335554_d_nat
            @ ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).

% Range_def
thf(fact_1060_Range__def,axiom,
    ( range_a_b
    = ( ^ [R2: set_Product_prod_a_b] :
          ( collect_b
          @ ( rangep_a_b
            @ ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).

% Range_def
thf(fact_1061_Range__def,axiom,
    ( range_8045137746665472906_a_b_a
    = ( ^ [R2: set_Pr8226792695654363348_a_b_a] :
          ( collec6734253767060832316_a_b_a
          @ ( rangep7418344918360899024_a_b_a
            @ ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).

% Range_def
thf(fact_1062_Range__def,axiom,
    ( range_5113499372716530458_a_b_a
    = ( ^ [R2: set_Pr2043227878544749668_a_b_a] :
          ( collec6734253767060832316_a_b_a
          @ ( rangep254816601053035104_a_b_a
            @ ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).

% Range_def
thf(fact_1063_Range__def,axiom,
    ( range_5462226226884687280_a_b_a
    = ( ^ [R2: set_Pr2084289749507318346_a_b_a] :
          ( collec6734253767060832316_a_b_a
          @ ( rangep1364177768494743018_a_b_a
            @ ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).

% Range_def
thf(fact_1064_bot__empty__eq2,axiom,
    ( bot_bo4560884378851987595_nat_o
    = ( ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ bot_bo534106183387690102_d_nat ) ) ) ).

% bot_empty_eq2
thf(fact_1065_bot__empty__eq2,axiom,
    ( bot_bot_a_b_o
    = ( ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ bot_bo3428410327501040244od_a_b ) ) ) ).

% bot_empty_eq2
thf(fact_1066_bot__empty__eq2,axiom,
    ( bot_bo8015583036438368799_b_a_o
    = ( ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ bot_bo4296644039882531648_a_b_a ) ) ) ).

% bot_empty_eq2
thf(fact_1067_bot__empty__eq2,axiom,
    ( bot_bo3470798565393788175_b_a_o
    = ( ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ bot_bo3809133976849826000_a_b_a ) ) ) ).

% bot_empty_eq2
thf(fact_1068_bot__empty__eq2,axiom,
    ( bot_bo7429559555918173987_b_a_o
    = ( ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ bot_bo890063853627817182_a_b_a ) ) ) ).

% bot_empty_eq2
thf(fact_1069_Rangep__Range__eq,axiom,
    ! [R3: set_Pr8583615357880026082_d_nat] :
      ( ( rangep7060683137087335554_d_nat
        @ ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ R3 ) )
      = ( ^ [X4: produc5456268773499131775_d_nat] : ( member8156885303890938774_d_nat @ X4 @ ( range_1428461333860462920_d_nat @ R3 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_1070_Rangep__Range__eq,axiom,
    ! [R3: set_Product_prod_a_b] :
      ( ( rangep_a_b
        @ ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ R3 ) )
      = ( ^ [X4: b] : ( member_b @ X4 @ ( range_a_b @ R3 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_1071_Rangep__Range__eq,axiom,
    ! [R3: set_Pr8226792695654363348_a_b_a] :
      ( ( rangep7418344918360899024_a_b_a
        @ ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ R3 ) )
      = ( ^ [X4: mappin2365780136655028071_a_b_a] : ( member2890181943093937918_a_b_a @ X4 @ ( range_8045137746665472906_a_b_a @ R3 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_1072_Rangep__Range__eq,axiom,
    ! [R3: set_Pr2043227878544749668_a_b_a] :
      ( ( rangep254816601053035104_a_b_a
        @ ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ R3 ) )
      = ( ^ [X4: mappin2365780136655028071_a_b_a] : ( member2890181943093937918_a_b_a @ X4 @ ( range_5113499372716530458_a_b_a @ R3 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_1073_Rangep__Range__eq,axiom,
    ! [R3: set_Pr2084289749507318346_a_b_a] :
      ( ( rangep1364177768494743018_a_b_a
        @ ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ R3 ) )
      = ( ^ [X4: mappin2365780136655028071_a_b_a] : ( member2890181943093937918_a_b_a @ X4 @ ( range_5462226226884687280_a_b_a @ R3 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_1074_ImageI,axiom,
    ! [A: a,B: produc5456268773499131775_d_nat,R3: set_Pr8583615357880026082_d_nat,A4: set_a] :
      ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ A @ B ) @ R3 )
     => ( ( member_a @ A @ A4 )
       => ( member8156885303890938774_d_nat @ B @ ( image_2058477355203106154_d_nat @ R3 @ A4 ) ) ) ) ).

% ImageI
thf(fact_1075_ImageI,axiom,
    ! [A: a,B: b,R3: set_Product_prod_a_b,A4: set_a] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B ) @ R3 )
     => ( ( member_a @ A @ A4 )
       => ( member_b @ B @ ( image_a_b @ R3 @ A4 ) ) ) ) ).

% ImageI
thf(fact_1076_ImageI,axiom,
    ! [A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a,R3: set_Pr8226792695654363348_a_b_a,A4: set_Pr8583615357880026082_d_nat] :
      ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ A @ B ) @ R3 )
     => ( ( member6046481508403577027_d_nat @ A @ A4 )
       => ( member2890181943093937918_a_b_a @ B @ ( image_2939044502918392552_a_b_a @ R3 @ A4 ) ) ) ) ).

% ImageI
thf(fact_1077_ImageI,axiom,
    ! [A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a,R3: set_Pr2043227878544749668_a_b_a,A4: set_li2034412094485292274_d_nat] :
      ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ A @ B ) @ R3 )
     => ( ( member136131075555046355_d_nat @ A @ A4 )
       => ( member2890181943093937918_a_b_a @ B @ ( image_7228979531701828472_a_b_a @ R3 @ A4 ) ) ) ) ).

% ImageI
thf(fact_1078_ImageI,axiom,
    ! [A: a,B: mappin2365780136655028071_a_b_a,R3: set_Pr2084289749507318346_a_b_a,A4: set_a] :
      ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ A @ B ) @ R3 )
     => ( ( member_a @ A @ A4 )
       => ( member2890181943093937918_a_b_a @ B @ ( image_7613109983325919954_a_b_a @ R3 @ A4 ) ) ) ) ).

% ImageI
thf(fact_1079_Image__singleton__iff,axiom,
    ! [B: produc5456268773499131775_d_nat,R3: set_Pr8583615357880026082_d_nat,A: a] :
      ( ( member8156885303890938774_d_nat @ B @ ( image_2058477355203106154_d_nat @ R3 @ ( insert_a2 @ A @ bot_bot_set_a ) ) )
      = ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ A @ B ) @ R3 ) ) ).

% Image_singleton_iff
thf(fact_1080_Image__singleton__iff,axiom,
    ! [B: b,R3: set_Product_prod_a_b,A: a] :
      ( ( member_b @ B @ ( image_a_b @ R3 @ ( insert_a2 @ A @ bot_bot_set_a ) ) )
      = ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B ) @ R3 ) ) ).

% Image_singleton_iff
thf(fact_1081_Image__singleton__iff,axiom,
    ! [B: mappin2365780136655028071_a_b_a,R3: set_Pr8226792695654363348_a_b_a,A: produc1990351791824763820_d_nat] :
      ( ( member2890181943093937918_a_b_a @ B @ ( image_2939044502918392552_a_b_a @ R3 @ ( insert934994597402771932_d_nat @ A @ bot_bo534106183387690102_d_nat ) ) )
      = ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ A @ B ) @ R3 ) ) ).

% Image_singleton_iff
thf(fact_1082_Image__singleton__iff,axiom,
    ! [B: mappin2365780136655028071_a_b_a,R3: set_Pr2043227878544749668_a_b_a,A: list_P6684558232516339260_d_nat] :
      ( ( member2890181943093937918_a_b_a @ B @ ( image_7228979531701828472_a_b_a @ R3 @ ( insert5227142572749542508_d_nat @ A @ bot_bo7515912283962571654_d_nat ) ) )
      = ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ A @ B ) @ R3 ) ) ).

% Image_singleton_iff
thf(fact_1083_Image__singleton__iff,axiom,
    ! [B: mappin2365780136655028071_a_b_a,R3: set_Pr2084289749507318346_a_b_a,A: a] :
      ( ( member2890181943093937918_a_b_a @ B @ ( image_7613109983325919954_a_b_a @ R3 @ ( insert_a2 @ A @ bot_bot_set_a ) ) )
      = ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ A @ B ) @ R3 ) ) ).

% Image_singleton_iff
thf(fact_1084_ImageE,axiom,
    ! [B: produc5456268773499131775_d_nat,R3: set_Pr8583615357880026082_d_nat,A4: set_a] :
      ( ( member8156885303890938774_d_nat @ B @ ( image_2058477355203106154_d_nat @ R3 @ A4 ) )
     => ~ ! [X3: a] :
            ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X3 @ B ) @ R3 )
           => ~ ( member_a @ X3 @ A4 ) ) ) ).

% ImageE
thf(fact_1085_ImageE,axiom,
    ! [B: b,R3: set_Product_prod_a_b,A4: set_a] :
      ( ( member_b @ B @ ( image_a_b @ R3 @ A4 ) )
     => ~ ! [X3: a] :
            ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X3 @ B ) @ R3 )
           => ~ ( member_a @ X3 @ A4 ) ) ) ).

% ImageE
thf(fact_1086_ImageE,axiom,
    ! [B: mappin2365780136655028071_a_b_a,R3: set_Pr8226792695654363348_a_b_a,A4: set_Pr8583615357880026082_d_nat] :
      ( ( member2890181943093937918_a_b_a @ B @ ( image_2939044502918392552_a_b_a @ R3 @ A4 ) )
     => ~ ! [X3: produc1990351791824763820_d_nat] :
            ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X3 @ B ) @ R3 )
           => ~ ( member6046481508403577027_d_nat @ X3 @ A4 ) ) ) ).

% ImageE
thf(fact_1087_ImageE,axiom,
    ! [B: mappin2365780136655028071_a_b_a,R3: set_Pr2043227878544749668_a_b_a,A4: set_li2034412094485292274_d_nat] :
      ( ( member2890181943093937918_a_b_a @ B @ ( image_7228979531701828472_a_b_a @ R3 @ A4 ) )
     => ~ ! [X3: list_P6684558232516339260_d_nat] :
            ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X3 @ B ) @ R3 )
           => ~ ( member136131075555046355_d_nat @ X3 @ A4 ) ) ) ).

% ImageE
thf(fact_1088_ImageE,axiom,
    ! [B: mappin2365780136655028071_a_b_a,R3: set_Pr2084289749507318346_a_b_a,A4: set_a] :
      ( ( member2890181943093937918_a_b_a @ B @ ( image_7613109983325919954_a_b_a @ R3 @ A4 ) )
     => ~ ! [X3: a] :
            ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X3 @ B ) @ R3 )
           => ~ ( member_a @ X3 @ A4 ) ) ) ).

% ImageE
thf(fact_1089_Image__iff,axiom,
    ! [B: produc5456268773499131775_d_nat,R3: set_Pr8583615357880026082_d_nat,A4: set_a] :
      ( ( member8156885303890938774_d_nat @ B @ ( image_2058477355203106154_d_nat @ R3 @ A4 ) )
      = ( ? [X4: a] :
            ( ( member_a @ X4 @ A4 )
            & ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ B ) @ R3 ) ) ) ) ).

% Image_iff
thf(fact_1090_Image__iff,axiom,
    ! [B: b,R3: set_Product_prod_a_b,A4: set_a] :
      ( ( member_b @ B @ ( image_a_b @ R3 @ A4 ) )
      = ( ? [X4: a] :
            ( ( member_a @ X4 @ A4 )
            & ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ B ) @ R3 ) ) ) ) ).

% Image_iff
thf(fact_1091_Image__iff,axiom,
    ! [B: mappin2365780136655028071_a_b_a,R3: set_Pr8226792695654363348_a_b_a,A4: set_Pr8583615357880026082_d_nat] :
      ( ( member2890181943093937918_a_b_a @ B @ ( image_2939044502918392552_a_b_a @ R3 @ A4 ) )
      = ( ? [X4: produc1990351791824763820_d_nat] :
            ( ( member6046481508403577027_d_nat @ X4 @ A4 )
            & ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ B ) @ R3 ) ) ) ) ).

% Image_iff
thf(fact_1092_Image__iff,axiom,
    ! [B: mappin2365780136655028071_a_b_a,R3: set_Pr2043227878544749668_a_b_a,A4: set_li2034412094485292274_d_nat] :
      ( ( member2890181943093937918_a_b_a @ B @ ( image_7228979531701828472_a_b_a @ R3 @ A4 ) )
      = ( ? [X4: list_P6684558232516339260_d_nat] :
            ( ( member136131075555046355_d_nat @ X4 @ A4 )
            & ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ B ) @ R3 ) ) ) ) ).

% Image_iff
thf(fact_1093_Image__iff,axiom,
    ! [B: mappin2365780136655028071_a_b_a,R3: set_Pr2084289749507318346_a_b_a,A4: set_a] :
      ( ( member2890181943093937918_a_b_a @ B @ ( image_7613109983325919954_a_b_a @ R3 @ A4 ) )
      = ( ? [X4: a] :
            ( ( member_a @ X4 @ A4 )
            & ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ B ) @ R3 ) ) ) ) ).

% Image_iff
thf(fact_1094_rev__ImageI,axiom,
    ! [A: a,A4: set_a,B: produc5456268773499131775_d_nat,R3: set_Pr8583615357880026082_d_nat] :
      ( ( member_a @ A @ A4 )
     => ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ A @ B ) @ R3 )
       => ( member8156885303890938774_d_nat @ B @ ( image_2058477355203106154_d_nat @ R3 @ A4 ) ) ) ) ).

% rev_ImageI
thf(fact_1095_rev__ImageI,axiom,
    ! [A: a,A4: set_a,B: b,R3: set_Product_prod_a_b] :
      ( ( member_a @ A @ A4 )
     => ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B ) @ R3 )
       => ( member_b @ B @ ( image_a_b @ R3 @ A4 ) ) ) ) ).

% rev_ImageI
thf(fact_1096_rev__ImageI,axiom,
    ! [A: produc1990351791824763820_d_nat,A4: set_Pr8583615357880026082_d_nat,B: mappin2365780136655028071_a_b_a,R3: set_Pr8226792695654363348_a_b_a] :
      ( ( member6046481508403577027_d_nat @ A @ A4 )
     => ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ A @ B ) @ R3 )
       => ( member2890181943093937918_a_b_a @ B @ ( image_2939044502918392552_a_b_a @ R3 @ A4 ) ) ) ) ).

% rev_ImageI
thf(fact_1097_rev__ImageI,axiom,
    ! [A: list_P6684558232516339260_d_nat,A4: set_li2034412094485292274_d_nat,B: mappin2365780136655028071_a_b_a,R3: set_Pr2043227878544749668_a_b_a] :
      ( ( member136131075555046355_d_nat @ A @ A4 )
     => ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ A @ B ) @ R3 )
       => ( member2890181943093937918_a_b_a @ B @ ( image_7228979531701828472_a_b_a @ R3 @ A4 ) ) ) ) ).

% rev_ImageI
thf(fact_1098_rev__ImageI,axiom,
    ! [A: a,A4: set_a,B: mappin2365780136655028071_a_b_a,R3: set_Pr2084289749507318346_a_b_a] :
      ( ( member_a @ A @ A4 )
     => ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ A @ B ) @ R3 )
       => ( member2890181943093937918_a_b_a @ B @ ( image_7613109983325919954_a_b_a @ R3 @ A4 ) ) ) ) ).

% rev_ImageI
thf(fact_1099_Image__singleton,axiom,
    ! [R3: set_Pr8583615357880026082_d_nat,A: a] :
      ( ( image_2058477355203106154_d_nat @ R3 @ ( insert_a2 @ A @ bot_bot_set_a ) )
      = ( collec4764653592595650004_d_nat
        @ ^ [B4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ A @ B4 ) @ R3 ) ) ) ).

% Image_singleton
thf(fact_1100_Image__singleton,axiom,
    ! [R3: set_Product_prod_a_b,A: a] :
      ( ( image_a_b @ R3 @ ( insert_a2 @ A @ bot_bot_set_a ) )
      = ( collect_b
        @ ^ [B4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B4 ) @ R3 ) ) ) ).

% Image_singleton
thf(fact_1101_Image__singleton,axiom,
    ! [R3: set_Pr8226792695654363348_a_b_a,A: produc1990351791824763820_d_nat] :
      ( ( image_2939044502918392552_a_b_a @ R3 @ ( insert934994597402771932_d_nat @ A @ bot_bo534106183387690102_d_nat ) )
      = ( collec6734253767060832316_a_b_a
        @ ^ [B4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ A @ B4 ) @ R3 ) ) ) ).

% Image_singleton
thf(fact_1102_Image__singleton,axiom,
    ! [R3: set_Pr2043227878544749668_a_b_a,A: list_P6684558232516339260_d_nat] :
      ( ( image_7228979531701828472_a_b_a @ R3 @ ( insert5227142572749542508_d_nat @ A @ bot_bo7515912283962571654_d_nat ) )
      = ( collec6734253767060832316_a_b_a
        @ ^ [B4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ A @ B4 ) @ R3 ) ) ) ).

% Image_singleton
thf(fact_1103_Image__singleton,axiom,
    ! [R3: set_Pr2084289749507318346_a_b_a,A: a] :
      ( ( image_7613109983325919954_a_b_a @ R3 @ ( insert_a2 @ A @ bot_bot_set_a ) )
      = ( collec6734253767060832316_a_b_a
        @ ^ [B4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ A @ B4 ) @ R3 ) ) ) ).

% Image_singleton
thf(fact_1104_prod__set__simps_I2_J,axiom,
    ! [X: a,Y5: produc5456268773499131775_d_nat] :
      ( ( basic_4433336097992580600_d_nat @ ( produc5865182755557626534_d_nat @ X @ Y5 ) )
      = ( insert8628257856670502319_d_nat @ Y5 @ bot_bo6373193931821895113_d_nat ) ) ).

% prod_set_simps(2)
thf(fact_1105_prod__set__simps_I2_J,axiom,
    ! [X: a,Y5: b] :
      ( ( basic_snds_a_b @ ( product_Pair_a_b @ X @ Y5 ) )
      = ( insert_b2 @ Y5 @ bot_bot_set_b ) ) ).

% prod_set_simps(2)
thf(fact_1106_prod__set__simps_I2_J,axiom,
    ! [X: produc1990351791824763820_d_nat,Y5: mappin2365780136655028071_a_b_a] :
      ( ( basic_5729739091838978778_a_b_a @ ( produc1681437058057156268_a_b_a @ X @ Y5 ) )
      = ( insert2272198572858264983_a_b_a @ Y5 @ bot_bo3815340846421858609_a_b_a ) ) ).

% prod_set_simps(2)
thf(fact_1107_prod__set__simps_I2_J,axiom,
    ! [X: list_P6684558232516339260_d_nat,Y5: mappin2365780136655028071_a_b_a] :
      ( ( basic_3021118406664780394_a_b_a @ ( produc2616710729536263484_a_b_a @ X @ Y5 ) )
      = ( insert2272198572858264983_a_b_a @ Y5 @ bot_bo3815340846421858609_a_b_a ) ) ).

% prod_set_simps(2)
thf(fact_1108_prod__set__simps_I2_J,axiom,
    ! [X: a,Y5: mappin2365780136655028071_a_b_a] :
      ( ( basic_2095140019604224608_a_b_a @ ( produc1583556749610718734_a_b_a @ X @ Y5 ) )
      = ( insert2272198572858264983_a_b_a @ Y5 @ bot_bo3815340846421858609_a_b_a ) ) ).

% prod_set_simps(2)
thf(fact_1109_pair__imageI,axiom,
    ! [A: a,B: produc5456268773499131775_d_nat,A4: set_Pr8583615357880026082_d_nat,F: a > produc5456268773499131775_d_nat > produc1990351791824763820_d_nat] :
      ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ A @ B ) @ A4 )
     => ( member6046481508403577027_d_nat @ ( F @ A @ B ) @ ( image_9040144901271416031_d_nat @ ( produc5272039204407661311_d_nat @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_1110_pair__imageI,axiom,
    ! [A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a,A4: set_Pr8226792695654363348_a_b_a,F: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a] :
      ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ A @ B ) @ A4 )
     => ( member4296969940938992029_a_b_a @ ( F @ A @ B ) @ ( image_3100555288041829637_a_b_a @ ( produc5613835436409813491_a_b_a @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_1111_pair__imageI,axiom,
    ! [A: a,B: produc5456268773499131775_d_nat,A4: set_Pr8583615357880026082_d_nat,F: a > produc5456268773499131775_d_nat > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a] :
      ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ A @ B ) @ A4 )
     => ( member4644490972414488365_a_b_a @ ( F @ A @ B ) @ ( image_4015665751426257425_a_b_a @ ( produc5638855421663032305_a_b_a @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_1112_pair__imageI,axiom,
    ! [A: a,B: mappin2365780136655028071_a_b_a,A4: set_Pr2084289749507318346_a_b_a,F: a > mappin2365780136655028071_a_b_a > produc6343861673719254644_a_b_a] :
      ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ A @ B ) @ A4 )
     => ( member4296969940938992029_a_b_a @ ( F @ A @ B ) @ ( image_8807444094480470169_a_b_a @ ( produc452744236373541113_a_b_a @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_1113_pair__imageI,axiom,
    ! [A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a,A4: set_Pr2043227878544749668_a_b_a,F: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a > produc2062766750192416900_a_b_a] :
      ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ A @ B ) @ A4 )
     => ( member50691962224995117_a_b_a @ ( F @ A @ B ) @ ( image_5843625383113718149_a_b_a @ ( produc3256888519411912691_a_b_a @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_1114_pair__imageI,axiom,
    ! [A: a,B: b,A4: set_Product_prod_a_b,F: a > b > product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B ) @ A4 )
     => ( member1426531481828664017od_a_b @ ( F @ A @ B ) @ ( image_3300603549555413765od_a_b @ ( produc8992199381948149691od_a_b @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_1115_sup__Un__eq2,axiom,
    ! [R: set_Pr8583615357880026082_d_nat,S: set_Pr8583615357880026082_d_nat] :
      ( ( sup_su1774166001322589987_nat_o
        @ ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ ( sup_su2239145386551726862_d_nat @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1116_sup__Un__eq2,axiom,
    ! [R: set_Product_prod_a_b,S: set_Product_prod_a_b] :
      ( ( sup_sup_a_b_o
        @ ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ R )
        @ ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ ( sup_su3119292821646003676od_a_b @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1117_sup__Un__eq2,axiom,
    ! [R: set_Pr8226792695654363348_a_b_a,S: set_Pr8226792695654363348_a_b_a] :
      ( ( sup_su1557148664756432263_b_a_o
        @ ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ R )
        @ ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ ( sup_su6789812448000380584_a_b_a @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1118_sup__Un__eq2,axiom,
    ! [R: set_Pr2043227878544749668_a_b_a,S: set_Pr2043227878544749668_a_b_a] :
      ( ( sup_su9160792860843062391_b_a_o
        @ ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ R )
        @ ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ ( sup_su1472619756083714104_a_b_a @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1119_sup__Un__eq2,axiom,
    ! [R: set_Pr2084289749507318346_a_b_a,S: set_Pr2084289749507318346_a_b_a] :
      ( ( sup_su1019514403191247291_b_a_o
        @ ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ R )
        @ ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ ( sup_su7721301866805749622_a_b_a @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1120_insert__Times__insert,axiom,
    ! [A: a,A4: set_a,B: produc5456268773499131775_d_nat,B6: set_Pr7704721104634554677_d_nat] :
      ( ( produc6456947282657855029_d_nat @ ( insert_a2 @ A @ A4 )
        @ ^ [Uu: a] : ( insert8628257856670502319_d_nat @ B @ B6 ) )
      = ( insert934994597402771932_d_nat @ ( produc5865182755557626534_d_nat @ A @ B )
        @ ( sup_su2239145386551726862_d_nat
          @ ( produc6456947282657855029_d_nat @ A4
            @ ^ [Uu: a] : ( insert8628257856670502319_d_nat @ B @ B6 ) )
          @ ( produc6456947282657855029_d_nat @ ( insert_a2 @ A @ A4 )
            @ ^ [Uu: a] : B6 ) ) ) ) ).

% insert_Times_insert
thf(fact_1121_insert__Times__insert,axiom,
    ! [A: a,A4: set_a,B: b,B6: set_b] :
      ( ( product_Sigma_a_b @ ( insert_a2 @ A @ A4 )
        @ ^ [Uu: a] : ( insert_b2 @ B @ B6 ) )
      = ( insert4534936386344385144od_a_b @ ( product_Pair_a_b @ A @ B )
        @ ( sup_su3119292821646003676od_a_b
          @ ( product_Sigma_a_b @ A4
            @ ^ [Uu: a] : ( insert_b2 @ B @ B6 ) )
          @ ( product_Sigma_a_b @ ( insert_a2 @ A @ A4 )
            @ ^ [Uu: a] : B6 ) ) ) ) ).

% insert_Times_insert
thf(fact_1122_insert__Times__insert,axiom,
    ! [A: produc1990351791824763820_d_nat,A4: set_Pr8583615357880026082_d_nat,B: mappin2365780136655028071_a_b_a,B6: set_ma506406390364637853_a_b_a] :
      ( ( produc3232134635143574877_a_b_a @ ( insert934994597402771932_d_nat @ A @ A4 )
        @ ^ [Uu: produc1990351791824763820_d_nat] : ( insert2272198572858264983_a_b_a @ B @ B6 ) )
      = ( insert1602870736147291204_a_b_a @ ( produc1681437058057156268_a_b_a @ A @ B )
        @ ( sup_su6789812448000380584_a_b_a
          @ ( produc3232134635143574877_a_b_a @ A4
            @ ^ [Uu: produc1990351791824763820_d_nat] : ( insert2272198572858264983_a_b_a @ B @ B6 ) )
          @ ( produc3232134635143574877_a_b_a @ ( insert934994597402771932_d_nat @ A @ A4 )
            @ ^ [Uu: produc1990351791824763820_d_nat] : B6 ) ) ) ) ).

% insert_Times_insert
thf(fact_1123_insert__Times__insert,axiom,
    ! [A: list_P6684558232516339260_d_nat,A4: set_li2034412094485292274_d_nat,B: mappin2365780136655028071_a_b_a,B6: set_ma506406390364637853_a_b_a] :
      ( ( produc534304670606313325_a_b_a @ ( insert5227142572749542508_d_nat @ A @ A4 )
        @ ^ [Uu: list_P6684558232516339260_d_nat] : ( insert2272198572858264983_a_b_a @ B @ B6 ) )
      = ( insert6197680461883956820_a_b_a @ ( produc2616710729536263484_a_b_a @ A @ B )
        @ ( sup_su1472619756083714104_a_b_a
          @ ( produc534304670606313325_a_b_a @ A4
            @ ^ [Uu: list_P6684558232516339260_d_nat] : ( insert2272198572858264983_a_b_a @ B @ B6 ) )
          @ ( produc534304670606313325_a_b_a @ ( insert5227142572749542508_d_nat @ A @ A4 )
            @ ^ [Uu: list_P6684558232516339260_d_nat] : B6 ) ) ) ) ).

% insert_Times_insert
thf(fact_1124_insert__Times__insert,axiom,
    ! [A: a,A4: set_a,B: mappin2365780136655028071_a_b_a,B6: set_ma506406390364637853_a_b_a] :
      ( ( produc6940724223187229_a_b_a @ ( insert_a2 @ A @ A4 )
        @ ^ [Uu: a] : ( insert2272198572858264983_a_b_a @ B @ B6 ) )
      = ( insert1871085374464671428_a_b_a @ ( produc1583556749610718734_a_b_a @ A @ B )
        @ ( sup_su7721301866805749622_a_b_a
          @ ( produc6940724223187229_a_b_a @ A4
            @ ^ [Uu: a] : ( insert2272198572858264983_a_b_a @ B @ B6 ) )
          @ ( produc6940724223187229_a_b_a @ ( insert_a2 @ A @ A4 )
            @ ^ [Uu: a] : B6 ) ) ) ) ).

% insert_Times_insert
thf(fact_1125_mem__Sigma__iff,axiom,
    ! [A: a,B: produc5456268773499131775_d_nat,A4: set_a,B6: a > set_Pr7704721104634554677_d_nat] :
      ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ A @ B ) @ ( produc6456947282657855029_d_nat @ A4 @ B6 ) )
      = ( ( member_a @ A @ A4 )
        & ( member8156885303890938774_d_nat @ B @ ( B6 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1126_mem__Sigma__iff,axiom,
    ! [A: a,B: b,A4: set_a,B6: a > set_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B ) @ ( product_Sigma_a_b @ A4 @ B6 ) )
      = ( ( member_a @ A @ A4 )
        & ( member_b @ B @ ( B6 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1127_mem__Sigma__iff,axiom,
    ! [A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a,A4: set_Pr8583615357880026082_d_nat,B6: produc1990351791824763820_d_nat > set_ma506406390364637853_a_b_a] :
      ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ A @ B ) @ ( produc3232134635143574877_a_b_a @ A4 @ B6 ) )
      = ( ( member6046481508403577027_d_nat @ A @ A4 )
        & ( member2890181943093937918_a_b_a @ B @ ( B6 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1128_mem__Sigma__iff,axiom,
    ! [A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a,A4: set_li2034412094485292274_d_nat,B6: list_P6684558232516339260_d_nat > set_ma506406390364637853_a_b_a] :
      ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ A @ B ) @ ( produc534304670606313325_a_b_a @ A4 @ B6 ) )
      = ( ( member136131075555046355_d_nat @ A @ A4 )
        & ( member2890181943093937918_a_b_a @ B @ ( B6 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1129_mem__Sigma__iff,axiom,
    ! [A: a,B: mappin2365780136655028071_a_b_a,A4: set_a,B6: a > set_ma506406390364637853_a_b_a] :
      ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ A @ B ) @ ( produc6940724223187229_a_b_a @ A4 @ B6 ) )
      = ( ( member_a @ A @ A4 )
        & ( member2890181943093937918_a_b_a @ B @ ( B6 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1130_SigmaI,axiom,
    ! [A: a,A4: set_a,B: produc5456268773499131775_d_nat,B6: a > set_Pr7704721104634554677_d_nat] :
      ( ( member_a @ A @ A4 )
     => ( ( member8156885303890938774_d_nat @ B @ ( B6 @ A ) )
       => ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ A @ B ) @ ( produc6456947282657855029_d_nat @ A4 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_1131_SigmaI,axiom,
    ! [A: a,A4: set_a,B: b,B6: a > set_b] :
      ( ( member_a @ A @ A4 )
     => ( ( member_b @ B @ ( B6 @ A ) )
       => ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B ) @ ( product_Sigma_a_b @ A4 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_1132_SigmaI,axiom,
    ! [A: produc1990351791824763820_d_nat,A4: set_Pr8583615357880026082_d_nat,B: mappin2365780136655028071_a_b_a,B6: produc1990351791824763820_d_nat > set_ma506406390364637853_a_b_a] :
      ( ( member6046481508403577027_d_nat @ A @ A4 )
     => ( ( member2890181943093937918_a_b_a @ B @ ( B6 @ A ) )
       => ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ A @ B ) @ ( produc3232134635143574877_a_b_a @ A4 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_1133_SigmaI,axiom,
    ! [A: list_P6684558232516339260_d_nat,A4: set_li2034412094485292274_d_nat,B: mappin2365780136655028071_a_b_a,B6: list_P6684558232516339260_d_nat > set_ma506406390364637853_a_b_a] :
      ( ( member136131075555046355_d_nat @ A @ A4 )
     => ( ( member2890181943093937918_a_b_a @ B @ ( B6 @ A ) )
       => ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ A @ B ) @ ( produc534304670606313325_a_b_a @ A4 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_1134_SigmaI,axiom,
    ! [A: a,A4: set_a,B: mappin2365780136655028071_a_b_a,B6: a > set_ma506406390364637853_a_b_a] :
      ( ( member_a @ A @ A4 )
     => ( ( member2890181943093937918_a_b_a @ B @ ( B6 @ A ) )
       => ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ A @ B ) @ ( produc6940724223187229_a_b_a @ A4 @ B6 ) ) ) ) ).

% SigmaI
thf(fact_1135_SigmaE2,axiom,
    ! [A: a,B: produc5456268773499131775_d_nat,A4: set_a,B6: a > set_Pr7704721104634554677_d_nat] :
      ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ A @ B ) @ ( produc6456947282657855029_d_nat @ A4 @ B6 ) )
     => ~ ( ( member_a @ A @ A4 )
         => ~ ( member8156885303890938774_d_nat @ B @ ( B6 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1136_SigmaE2,axiom,
    ! [A: a,B: b,A4: set_a,B6: a > set_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B ) @ ( product_Sigma_a_b @ A4 @ B6 ) )
     => ~ ( ( member_a @ A @ A4 )
         => ~ ( member_b @ B @ ( B6 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1137_SigmaE2,axiom,
    ! [A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a,A4: set_Pr8583615357880026082_d_nat,B6: produc1990351791824763820_d_nat > set_ma506406390364637853_a_b_a] :
      ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ A @ B ) @ ( produc3232134635143574877_a_b_a @ A4 @ B6 ) )
     => ~ ( ( member6046481508403577027_d_nat @ A @ A4 )
         => ~ ( member2890181943093937918_a_b_a @ B @ ( B6 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1138_SigmaE2,axiom,
    ! [A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a,A4: set_li2034412094485292274_d_nat,B6: list_P6684558232516339260_d_nat > set_ma506406390364637853_a_b_a] :
      ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ A @ B ) @ ( produc534304670606313325_a_b_a @ A4 @ B6 ) )
     => ~ ( ( member136131075555046355_d_nat @ A @ A4 )
         => ~ ( member2890181943093937918_a_b_a @ B @ ( B6 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1139_SigmaE2,axiom,
    ! [A: a,B: mappin2365780136655028071_a_b_a,A4: set_a,B6: a > set_ma506406390364637853_a_b_a] :
      ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ A @ B ) @ ( produc6940724223187229_a_b_a @ A4 @ B6 ) )
     => ~ ( ( member_a @ A @ A4 )
         => ~ ( member2890181943093937918_a_b_a @ B @ ( B6 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1140_SigmaD2,axiom,
    ! [A: a,B: produc5456268773499131775_d_nat,A4: set_a,B6: a > set_Pr7704721104634554677_d_nat] :
      ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ A @ B ) @ ( produc6456947282657855029_d_nat @ A4 @ B6 ) )
     => ( member8156885303890938774_d_nat @ B @ ( B6 @ A ) ) ) ).

% SigmaD2
thf(fact_1141_SigmaD2,axiom,
    ! [A: a,B: b,A4: set_a,B6: a > set_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B ) @ ( product_Sigma_a_b @ A4 @ B6 ) )
     => ( member_b @ B @ ( B6 @ A ) ) ) ).

% SigmaD2
thf(fact_1142_SigmaD2,axiom,
    ! [A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a,A4: set_Pr8583615357880026082_d_nat,B6: produc1990351791824763820_d_nat > set_ma506406390364637853_a_b_a] :
      ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ A @ B ) @ ( produc3232134635143574877_a_b_a @ A4 @ B6 ) )
     => ( member2890181943093937918_a_b_a @ B @ ( B6 @ A ) ) ) ).

% SigmaD2
thf(fact_1143_SigmaD2,axiom,
    ! [A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a,A4: set_li2034412094485292274_d_nat,B6: list_P6684558232516339260_d_nat > set_ma506406390364637853_a_b_a] :
      ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ A @ B ) @ ( produc534304670606313325_a_b_a @ A4 @ B6 ) )
     => ( member2890181943093937918_a_b_a @ B @ ( B6 @ A ) ) ) ).

% SigmaD2
thf(fact_1144_SigmaD2,axiom,
    ! [A: a,B: mappin2365780136655028071_a_b_a,A4: set_a,B6: a > set_ma506406390364637853_a_b_a] :
      ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ A @ B ) @ ( produc6940724223187229_a_b_a @ A4 @ B6 ) )
     => ( member2890181943093937918_a_b_a @ B @ ( B6 @ A ) ) ) ).

% SigmaD2
thf(fact_1145_SigmaD1,axiom,
    ! [A: a,B: produc5456268773499131775_d_nat,A4: set_a,B6: a > set_Pr7704721104634554677_d_nat] :
      ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ A @ B ) @ ( produc6456947282657855029_d_nat @ A4 @ B6 ) )
     => ( member_a @ A @ A4 ) ) ).

% SigmaD1
thf(fact_1146_SigmaD1,axiom,
    ! [A: a,B: b,A4: set_a,B6: a > set_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A @ B ) @ ( product_Sigma_a_b @ A4 @ B6 ) )
     => ( member_a @ A @ A4 ) ) ).

% SigmaD1
thf(fact_1147_SigmaD1,axiom,
    ! [A: produc1990351791824763820_d_nat,B: mappin2365780136655028071_a_b_a,A4: set_Pr8583615357880026082_d_nat,B6: produc1990351791824763820_d_nat > set_ma506406390364637853_a_b_a] :
      ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ A @ B ) @ ( produc3232134635143574877_a_b_a @ A4 @ B6 ) )
     => ( member6046481508403577027_d_nat @ A @ A4 ) ) ).

% SigmaD1
thf(fact_1148_SigmaD1,axiom,
    ! [A: list_P6684558232516339260_d_nat,B: mappin2365780136655028071_a_b_a,A4: set_li2034412094485292274_d_nat,B6: list_P6684558232516339260_d_nat > set_ma506406390364637853_a_b_a] :
      ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ A @ B ) @ ( produc534304670606313325_a_b_a @ A4 @ B6 ) )
     => ( member136131075555046355_d_nat @ A @ A4 ) ) ).

% SigmaD1
thf(fact_1149_SigmaD1,axiom,
    ! [A: a,B: mappin2365780136655028071_a_b_a,A4: set_a,B6: a > set_ma506406390364637853_a_b_a] :
      ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ A @ B ) @ ( produc6940724223187229_a_b_a @ A4 @ B6 ) )
     => ( member_a @ A @ A4 ) ) ).

% SigmaD1
thf(fact_1150_SigmaE,axiom,
    ! [C2: produc1990351791824763820_d_nat,A4: set_a,B6: a > set_Pr7704721104634554677_d_nat] :
      ( ( member6046481508403577027_d_nat @ C2 @ ( produc6456947282657855029_d_nat @ A4 @ B6 ) )
     => ~ ! [X3: a] :
            ( ( member_a @ X3 @ A4 )
           => ! [Y3: produc5456268773499131775_d_nat] :
                ( ( member8156885303890938774_d_nat @ Y3 @ ( B6 @ X3 ) )
               => ( C2
                 != ( produc5865182755557626534_d_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_1151_SigmaE,axiom,
    ! [C2: product_prod_a_b,A4: set_a,B6: a > set_b] :
      ( ( member1426531481828664017od_a_b @ C2 @ ( product_Sigma_a_b @ A4 @ B6 ) )
     => ~ ! [X3: a] :
            ( ( member_a @ X3 @ A4 )
           => ! [Y3: b] :
                ( ( member_b @ Y3 @ ( B6 @ X3 ) )
               => ( C2
                 != ( product_Pair_a_b @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_1152_SigmaE,axiom,
    ! [C2: produc6343861673719254644_a_b_a,A4: set_Pr8583615357880026082_d_nat,B6: produc1990351791824763820_d_nat > set_ma506406390364637853_a_b_a] :
      ( ( member4296969940938992029_a_b_a @ C2 @ ( produc3232134635143574877_a_b_a @ A4 @ B6 ) )
     => ~ ! [X3: produc1990351791824763820_d_nat] :
            ( ( member6046481508403577027_d_nat @ X3 @ A4 )
           => ! [Y3: mappin2365780136655028071_a_b_a] :
                ( ( member2890181943093937918_a_b_a @ Y3 @ ( B6 @ X3 ) )
               => ( C2
                 != ( produc1681437058057156268_a_b_a @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_1153_SigmaE,axiom,
    ! [C2: produc2062766750192416900_a_b_a,A4: set_li2034412094485292274_d_nat,B6: list_P6684558232516339260_d_nat > set_ma506406390364637853_a_b_a] :
      ( ( member50691962224995117_a_b_a @ C2 @ ( produc534304670606313325_a_b_a @ A4 @ B6 ) )
     => ~ ! [X3: list_P6684558232516339260_d_nat] :
            ( ( member136131075555046355_d_nat @ X3 @ A4 )
           => ! [Y3: mappin2365780136655028071_a_b_a] :
                ( ( member2890181943093937918_a_b_a @ Y3 @ ( B6 @ X3 ) )
               => ( C2
                 != ( produc2616710729536263484_a_b_a @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_1154_SigmaE,axiom,
    ! [C2: produc7822699449245016212_a_b_a,A4: set_a,B6: a > set_ma506406390364637853_a_b_a] :
      ( ( member3694737009695996203_a_b_a @ C2 @ ( produc6940724223187229_a_b_a @ A4 @ B6 ) )
     => ~ ! [X3: a] :
            ( ( member_a @ X3 @ A4 )
           => ! [Y3: mappin2365780136655028071_a_b_a] :
                ( ( member2890181943093937918_a_b_a @ Y3 @ ( B6 @ X3 ) )
               => ( C2
                 != ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_1155_Sigma__def,axiom,
    ( produc6456947282657855029_d_nat
    = ( ^ [A7: set_a,B7: a > set_Pr7704721104634554677_d_nat] :
          ( comple1064892979059583925_d_nat
          @ ( image_4079803252418282267_d_nat
            @ ^ [X4: a] :
                ( comple1064892979059583925_d_nat
                @ ( image_7769734296291535208_d_nat
                  @ ^ [Y4: produc5456268773499131775_d_nat] : ( insert934994597402771932_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ bot_bo534106183387690102_d_nat )
                  @ ( B7 @ X4 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_1156_Sigma__def,axiom,
    ( product_Sigma_a_b
    = ( ^ [A7: set_a,B7: a > set_b] :
          ( comple8492713210738191477od_a_b
          @ ( image_4492544633037792655od_a_b
            @ ^ [X4: a] :
                ( comple8492713210738191477od_a_b
                @ ( image_5733669806535660750od_a_b
                  @ ^ [Y4: b] : ( insert4534936386344385144od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ bot_bo3428410327501040244od_a_b )
                  @ ( B7 @ X4 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_1157_Sigma__def,axiom,
    ( produc3232134635143574877_a_b_a
    = ( ^ [A7: set_Pr8583615357880026082_d_nat,B7: produc1990351791824763820_d_nat > set_ma506406390364637853_a_b_a] :
          ( comple1936190902934540865_a_b_a
          @ ( image_5085805805582056673_a_b_a
            @ ^ [X4: produc1990351791824763820_d_nat] :
                ( comple1936190902934540865_a_b_a
                @ ( image_4454330836619085030_a_b_a
                  @ ^ [Y4: mappin2365780136655028071_a_b_a] : ( insert1602870736147291204_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ bot_bo4296644039882531648_a_b_a )
                  @ ( B7 @ X4 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_1158_Sigma__def,axiom,
    ( produc534304670606313325_a_b_a
    = ( ^ [A7: set_li2034412094485292274_d_nat,B7: list_P6684558232516339260_d_nat > set_ma506406390364637853_a_b_a] :
          ( comple719183236107303761_a_b_a
          @ ( image_4525829845069750881_a_b_a
            @ ^ [X4: list_P6684558232516339260_d_nat] :
                ( comple719183236107303761_a_b_a
                @ ( image_873846514519218038_a_b_a
                  @ ^ [Y4: mappin2365780136655028071_a_b_a] : ( insert6197680461883956820_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ bot_bo3809133976849826000_a_b_a )
                  @ ( B7 @ X4 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_1159_Sigma__def,axiom,
    ( produc6940724223187229_a_b_a
    = ( ^ [A7: set_a,B7: a > set_ma506406390364637853_a_b_a] :
          ( comple6773161314375097245_a_b_a
          @ ( image_3280393231850501635_a_b_a
            @ ^ [X4: a] :
                ( comple6773161314375097245_a_b_a
                @ ( image_7977788677304419768_a_b_a
                  @ ^ [Y4: mappin2365780136655028071_a_b_a] : ( insert1871085374464671428_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ bot_bo890063853627817182_a_b_a )
                  @ ( B7 @ X4 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_1160_swap__product,axiom,
    ! [A4: set_Pr7704721104634554677_d_nat,B6: set_a] :
      ( ( image_3560925789152379495_d_nat
        @ ( produc5894518052405473087_d_nat
          @ ^ [I: produc5456268773499131775_d_nat,J: a] : ( produc5865182755557626534_d_nat @ J @ I ) )
        @ ( produc1760853684164264565_nat_a @ A4
          @ ^ [Uu: produc5456268773499131775_d_nat] : B6 ) )
      = ( produc6456947282657855029_d_nat @ B6
        @ ^ [Uu: a] : A4 ) ) ).

% swap_product
thf(fact_1161_swap__product,axiom,
    ! [A4: set_b,B6: set_a] :
      ( ( image_8576602756132944327od_a_b
        @ ( produc4348216232050026237od_a_b
          @ ^ [I: b,J: a] : ( product_Pair_a_b @ J @ I ) )
        @ ( product_Sigma_b_a @ A4
          @ ^ [Uu: b] : B6 ) )
      = ( product_Sigma_a_b @ B6
        @ ^ [Uu: a] : A4 ) ) ).

% swap_product
thf(fact_1162_swap__product,axiom,
    ! [A4: set_ma506406390364637853_a_b_a,B6: set_Pr8583615357880026082_d_nat] :
      ( ( image_8374183271727160965_a_b_a
        @ ( produc3170874359442458355_a_b_a
          @ ^ [I: mappin2365780136655028071_a_b_a,J: produc1990351791824763820_d_nat] : ( produc1681437058057156268_a_b_a @ J @ I ) )
        @ ( produc7933817083176790109_d_nat @ A4
          @ ^ [Uu: mappin2365780136655028071_a_b_a] : B6 ) )
      = ( produc3232134635143574877_a_b_a @ B6
        @ ^ [Uu: produc1990351791824763820_d_nat] : A4 ) ) ).

% swap_product
thf(fact_1163_swap__product,axiom,
    ! [A4: set_ma506406390364637853_a_b_a,B6: set_li2034412094485292274_d_nat] :
      ( ( image_7372413891761560837_a_b_a
        @ ( produc5658256351460266227_a_b_a
          @ ^ [I: mappin2365780136655028071_a_b_a,J: list_P6684558232516339260_d_nat] : ( produc2616710729536263484_a_b_a @ J @ I ) )
        @ ( produc3833965954989344365_d_nat @ A4
          @ ^ [Uu: mappin2365780136655028071_a_b_a] : B6 ) )
      = ( produc534304670606313325_a_b_a @ B6
        @ ^ [Uu: list_P6684558232516339260_d_nat] : A4 ) ) ).

% swap_product
thf(fact_1164_swap__product,axiom,
    ! [A4: set_ma506406390364637853_a_b_a,B6: set_a] :
      ( ( image_7492227643763088743_a_b_a
        @ ( produc7265838118961918399_a_b_a
          @ ^ [I: mappin2365780136655028071_a_b_a,J: a] : ( produc1583556749610718734_a_b_a @ J @ I ) )
        @ ( produc941370498625271821_b_a_a @ A4
          @ ^ [Uu: mappin2365780136655028071_a_b_a] : B6 ) )
      = ( produc6940724223187229_a_b_a @ B6
        @ ^ [Uu: a] : A4 ) ) ).

% swap_product
thf(fact_1165_image__paired__Times,axiom,
    ! [F: a > a,G: produc5456268773499131775_d_nat > produc5456268773499131775_d_nat,A4: set_a,B6: set_Pr7704721104634554677_d_nat] :
      ( ( image_9040144901271416031_d_nat
        @ ( produc5272039204407661311_d_nat
          @ ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( produc5865182755557626534_d_nat @ ( F @ X4 ) @ ( G @ Y4 ) ) )
        @ ( produc6456947282657855029_d_nat @ A4
          @ ^ [Uu: a] : B6 ) )
      = ( produc6456947282657855029_d_nat @ ( image_a_a @ F @ A4 )
        @ ^ [Uu: a] : ( image_4410307165664250629_d_nat @ G @ B6 ) ) ) ).

% image_paired_Times
thf(fact_1166_image__paired__Times,axiom,
    ! [F: produc1990351791824763820_d_nat > produc1990351791824763820_d_nat,G: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,A4: set_Pr8583615357880026082_d_nat,B6: set_ma506406390364637853_a_b_a] :
      ( ( image_3100555288041829637_a_b_a
        @ ( produc5613835436409813491_a_b_a
          @ ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( produc1681437058057156268_a_b_a @ ( F @ X4 ) @ ( G @ Y4 ) ) )
        @ ( produc3232134635143574877_a_b_a @ A4
          @ ^ [Uu: produc1990351791824763820_d_nat] : B6 ) )
      = ( produc3232134635143574877_a_b_a @ ( image_9040144901271416031_d_nat @ F @ A4 )
        @ ^ [Uu: produc1990351791824763820_d_nat] : ( image_8981982570873075925_a_b_a @ G @ B6 ) ) ) ).

% image_paired_Times
thf(fact_1167_image__paired__Times,axiom,
    ! [F: a > produc1990351791824763820_d_nat,G: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,A4: set_a,B6: set_ma506406390364637853_a_b_a] :
      ( ( image_8807444094480470169_a_b_a
        @ ( produc452744236373541113_a_b_a
          @ ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( produc1681437058057156268_a_b_a @ ( F @ X4 ) @ ( G @ Y4 ) ) )
        @ ( produc6940724223187229_a_b_a @ A4
          @ ^ [Uu: a] : B6 ) )
      = ( produc3232134635143574877_a_b_a @ ( image_3542610352415022821_d_nat @ F @ A4 )
        @ ^ [Uu: produc1990351791824763820_d_nat] : ( image_8981982570873075925_a_b_a @ G @ B6 ) ) ) ).

% image_paired_Times
thf(fact_1168_image__paired__Times,axiom,
    ! [F: list_P6684558232516339260_d_nat > list_P6684558232516339260_d_nat,G: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,A4: set_li2034412094485292274_d_nat,B6: set_ma506406390364637853_a_b_a] :
      ( ( image_5843625383113718149_a_b_a
        @ ( produc3256888519411912691_a_b_a
          @ ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( produc2616710729536263484_a_b_a @ ( F @ X4 ) @ ( G @ Y4 ) ) )
        @ ( produc534304670606313325_a_b_a @ A4
          @ ^ [Uu: list_P6684558232516339260_d_nat] : B6 ) )
      = ( produc534304670606313325_a_b_a @ ( image_939471793936896383_d_nat @ F @ A4 )
        @ ^ [Uu: list_P6684558232516339260_d_nat] : ( image_8981982570873075925_a_b_a @ G @ B6 ) ) ) ).

% image_paired_Times
thf(fact_1169_image__paired__Times,axiom,
    ! [F: a > a,G: b > b,A4: set_a,B6: set_b] :
      ( ( image_3300603549555413765od_a_b
        @ ( produc8992199381948149691od_a_b
          @ ^ [X4: a,Y4: b] : ( product_Pair_a_b @ ( F @ X4 ) @ ( G @ Y4 ) ) )
        @ ( product_Sigma_a_b @ A4
          @ ^ [Uu: a] : B6 ) )
      = ( product_Sigma_a_b @ ( image_a_a @ F @ A4 )
        @ ^ [Uu: a] : ( image_b_b @ G @ B6 ) ) ) ).

% image_paired_Times
thf(fact_1170_SUP__Sup__eq2,axiom,
    ! [S: set_se3561969970571257112_d_nat] :
      ( ( comple112106235062927818_nat_o
        @ ( image_8176980452443277408_nat_o
          @ ^ [I: set_Pr8583615357880026082_d_nat,X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ I )
          @ S ) )
      = ( ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ ( comple1064892979059583925_d_nat @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_1171_SUP__Sup__eq2,axiom,
    ! [S: set_se3214616458824132456od_a_b] :
      ( ( comple2905715993080428826_a_b_o
        @ ( image_8629710542337338282_a_b_o
          @ ^ [I: set_Product_prod_a_b,X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ I )
          @ S ) )
      = ( ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ ( comple8492713210738191477od_a_b @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_1172_SUP__Sup__eq2,axiom,
    ! [S: set_se1240140977111469876_a_b_a] :
      ( ( comple2001304245037778720_b_a_o
        @ ( image_1759233996081757668_b_a_o
          @ ^ [I: set_Pr8226792695654363348_a_b_a,X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ I )
          @ S ) )
      = ( ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ ( comple1936190902934540865_a_b_a @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_1173_SUP__Sup__eq2,axiom,
    ! [S: set_se6657180468042755140_a_b_a] :
      ( ( comple4560970680813448592_b_a_o
        @ ( image_8355691765331406788_b_a_o
          @ ^ [I: set_Pr2043227878544749668_a_b_a,X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ I )
          @ S ) )
      = ( ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ ( comple719183236107303761_a_b_a @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_1174_SUP__Sup__eq2,axiom,
    ! [S: set_se7745462473697426688_a_b_a] :
      ( ( comple2234735975415836642_b_a_o
        @ ( image_741601757749928672_b_a_o
          @ ^ [I: set_Pr2084289749507318346_a_b_a,X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ I )
          @ S ) )
      = ( ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ ( comple6773161314375097245_a_b_a @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_1175_Sup__SUP__eq2,axiom,
    ( comple112106235062927818_nat_o
    = ( ^ [S11: set_a_7514012873179435181_nat_o,X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ ( comple1064892979059583925_d_nat @ ( image_3130169668052712384_d_nat @ collec3895662986305461505_d_nat @ ( image_5276271149719299749_nat_o @ produc1905287380190322693_nat_o @ S11 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_1176_Sup__SUP__eq2,axiom,
    ( comple2905715993080428826_a_b_o
    = ( ^ [S11: set_a_b_o,X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ ( comple8492713210738191477od_a_b @ ( image_5056655251382583084od_a_b @ collec3336397801687681299od_a_b @ ( image_5956505685088102973_a_b_o @ produc3537405659489547051_a_b_o @ S11 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_1177_Sup__SUP__eq2,axiom,
    ( comple2001304245037778720_b_a_o
    = ( ^ [S11: set_Pr9119034101783098515_b_a_o,X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ ( comple1936190902934540865_a_b_a @ ( image_1611519598313298116_a_b_a @ collec4442262398806403551_a_b_a @ ( image_5007618793744694903_b_a_o @ produc6407700812349938327_b_a_o @ S11 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_1178_Sup__SUP__eq2,axiom,
    ( comple4560970680813448592_b_a_o
    = ( ^ [S11: set_li6280422973213826819_b_a_o,X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ ( comple719183236107303761_a_b_a @ ( image_4387920688708098276_a_b_a @ collec2527315220090610287_a_b_a @ ( image_2537138341275166167_b_a_o @ produc6164275942970966535_b_a_o @ S11 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_1179_Sup__SUP__eq2,axiom,
    ( comple2234735975415836642_b_a_o
    = ( ^ [S11: set_a_7235201166916292293_b_a_o,X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ ( comple6773161314375097245_a_b_a @ ( image_4033343064979391424_a_b_a @ collec8037893338069451881_a_b_a @ ( image_3866804513976791973_b_a_o @ produc2053338206788791453_b_a_o @ S11 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_1180_subrelI,axiom,
    ! [R3: set_Pr8583615357880026082_d_nat,S3: set_Pr8583615357880026082_d_nat] :
      ( ! [X3: a,Y3: produc5456268773499131775_d_nat] :
          ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X3 @ Y3 ) @ R3 )
         => ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le3404350072972973122_d_nat @ R3 @ S3 ) ) ).

% subrelI
thf(fact_1181_subrelI,axiom,
    ! [R3: set_Product_prod_a_b,S3: set_Product_prod_a_b] :
      ( ! [X3: a,Y3: b] :
          ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X3 @ Y3 ) @ R3 )
         => ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le817736998455962536od_a_b @ R3 @ S3 ) ) ).

% subrelI
thf(fact_1182_subrelI,axiom,
    ! [R3: set_Pr8226792695654363348_a_b_a,S3: set_Pr8226792695654363348_a_b_a] :
      ( ! [X3: produc1990351791824763820_d_nat,Y3: mappin2365780136655028071_a_b_a] :
          ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X3 @ Y3 ) @ R3 )
         => ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le5314967415308617844_a_b_a @ R3 @ S3 ) ) ).

% subrelI
thf(fact_1183_subrelI,axiom,
    ! [R3: set_Pr2043227878544749668_a_b_a,S3: set_Pr2043227878544749668_a_b_a] :
      ( ! [X3: list_P6684558232516339260_d_nat,Y3: mappin2365780136655028071_a_b_a] :
          ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X3 @ Y3 ) @ R3 )
         => ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le7668720226886925316_a_b_a @ R3 @ S3 ) ) ).

% subrelI
thf(fact_1184_subrelI,axiom,
    ! [R3: set_Pr2084289749507318346_a_b_a,S3: set_Pr2084289749507318346_a_b_a] :
      ( ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
          ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) @ R3 )
         => ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) @ S3 ) )
     => ( ord_le7913445190064841386_a_b_a @ R3 @ S3 ) ) ).

% subrelI
thf(fact_1185_pred__subset__eq2,axiom,
    ! [R: set_Pr8583615357880026082_d_nat,S: set_Pr8583615357880026082_d_nat] :
      ( ( ord_le4807968413125245015_nat_o
        @ ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ S ) )
      = ( ord_le3404350072972973122_d_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_1186_pred__subset__eq2,axiom,
    ! [R: set_Product_prod_a_b,S: set_Product_prod_a_b] :
      ( ( ord_less_eq_a_b_o
        @ ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ R )
        @ ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ S ) )
      = ( ord_le817736998455962536od_a_b @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_1187_pred__subset__eq2,axiom,
    ! [R: set_Pr8226792695654363348_a_b_a,S: set_Pr8226792695654363348_a_b_a] :
      ( ( ord_le6298380606558187347_b_a_o
        @ ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ R )
        @ ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ S ) )
      = ( ord_le5314967415308617844_a_b_a @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_1188_pred__subset__eq2,axiom,
    ! [R: set_Pr2043227878544749668_a_b_a,S: set_Pr2043227878544749668_a_b_a] :
      ( ( ord_le8406588954588946499_b_a_o
        @ ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ R )
        @ ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ S ) )
      = ( ord_le7668720226886925316_a_b_a @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_1189_pred__subset__eq2,axiom,
    ! [R: set_Pr2084289749507318346_a_b_a,S: set_Pr2084289749507318346_a_b_a] :
      ( ( ord_le4941806563921773807_b_a_o
        @ ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ R )
        @ ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ S ) )
      = ( ord_le7913445190064841386_a_b_a @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_1190_GrD2,axiom,
    ! [X: a,Fx: produc5456268773499131775_d_nat,A4: set_a,F: a > produc5456268773499131775_d_nat] :
      ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X @ Fx ) @ ( bNF_Gr316990760905371227_d_nat @ A4 @ F ) )
     => ( ( F @ X )
        = Fx ) ) ).

% GrD2
thf(fact_1191_GrD2,axiom,
    ! [X: a,Fx: b,A4: set_a,F: a > b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X @ Fx ) @ ( bNF_Gr_a_b @ A4 @ F ) )
     => ( ( F @ X )
        = Fx ) ) ).

% GrD2
thf(fact_1192_GrD2,axiom,
    ! [X: produc1990351791824763820_d_nat,Fx: mappin2365780136655028071_a_b_a,A4: set_Pr8583615357880026082_d_nat,F: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a] :
      ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X @ Fx ) @ ( bNF_Gr4635658228676329399_a_b_a @ A4 @ F ) )
     => ( ( F @ X )
        = Fx ) ) ).

% GrD2
thf(fact_1193_GrD2,axiom,
    ! [X: list_P6684558232516339260_d_nat,Fx: mappin2365780136655028071_a_b_a,A4: set_li2034412094485292274_d_nat,F: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a] :
      ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X @ Fx ) @ ( bNF_Gr7807866264169389255_a_b_a @ A4 @ F ) )
     => ( ( F @ X )
        = Fx ) ) ).

% GrD2
thf(fact_1194_GrD2,axiom,
    ! [X: a,Fx: mappin2365780136655028071_a_b_a,A4: set_a,F: a > mappin2365780136655028071_a_b_a] :
      ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X @ Fx ) @ ( bNF_Gr4216663935370044483_a_b_a @ A4 @ F ) )
     => ( ( F @ X )
        = Fx ) ) ).

% GrD2
thf(fact_1195_GrD1,axiom,
    ! [X: a,Fx: produc5456268773499131775_d_nat,A4: set_a,F: a > produc5456268773499131775_d_nat] :
      ( ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X @ Fx ) @ ( bNF_Gr316990760905371227_d_nat @ A4 @ F ) )
     => ( member_a @ X @ A4 ) ) ).

% GrD1
thf(fact_1196_GrD1,axiom,
    ! [X: a,Fx: b,A4: set_a,F: a > b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X @ Fx ) @ ( bNF_Gr_a_b @ A4 @ F ) )
     => ( member_a @ X @ A4 ) ) ).

% GrD1
thf(fact_1197_GrD1,axiom,
    ! [X: produc1990351791824763820_d_nat,Fx: mappin2365780136655028071_a_b_a,A4: set_Pr8583615357880026082_d_nat,F: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a] :
      ( ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X @ Fx ) @ ( bNF_Gr4635658228676329399_a_b_a @ A4 @ F ) )
     => ( member6046481508403577027_d_nat @ X @ A4 ) ) ).

% GrD1
thf(fact_1198_GrD1,axiom,
    ! [X: list_P6684558232516339260_d_nat,Fx: mappin2365780136655028071_a_b_a,A4: set_li2034412094485292274_d_nat,F: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a] :
      ( ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X @ Fx ) @ ( bNF_Gr7807866264169389255_a_b_a @ A4 @ F ) )
     => ( member136131075555046355_d_nat @ X @ A4 ) ) ).

% GrD1
thf(fact_1199_GrD1,axiom,
    ! [X: a,Fx: mappin2365780136655028071_a_b_a,A4: set_a,F: a > mappin2365780136655028071_a_b_a] :
      ( ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X @ Fx ) @ ( bNF_Gr4216663935370044483_a_b_a @ A4 @ F ) )
     => ( member_a @ X @ A4 ) ) ).

% GrD1
thf(fact_1200_top__empty__eq2,axiom,
    ( top_to1872082848596987559_nat_o
    = ( ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ top_to9179421613355420306_d_nat ) ) ) ).

% top_empty_eq2
thf(fact_1201_top__empty__eq2,axiom,
    ( top_top_a_b_o
    = ( ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ top_to8134405472303993176od_a_b ) ) ) ).

% top_empty_eq2
thf(fact_1202_top__empty__eq2,axiom,
    ( top_to3417851538219541763_b_a_o
    = ( ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ top_to9182889407010954788_a_b_a ) ) ) ).

% top_empty_eq2
thf(fact_1203_top__empty__eq2,axiom,
    ( top_to1244882801596923379_b_a_o
    = ( ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ top_to520280237134248372_a_b_a ) ) ) ).

% top_empty_eq2
thf(fact_1204_top__empty__eq2,axiom,
    ( top_to6739199018942169919_b_a_o
    = ( ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ top_to5556156715448795386_a_b_a ) ) ) ).

% top_empty_eq2
thf(fact_1205_inf__Int__eq2,axiom,
    ! [R: set_Pr8583615357880026082_d_nat,S: set_Pr8583615357880026082_d_nat] :
      ( ( inf_in4633533406897155465_nat_o
        @ ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ R )
        @ ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ ( inf_in5260450986781187444_d_nat @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1206_inf__Int__eq2,axiom,
    ! [R: set_Product_prod_a_b,S: set_Product_prod_a_b] :
      ( ( inf_inf_a_b_o
        @ ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ R )
        @ ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ ( inf_in8976041639890736118od_a_b @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1207_inf__Int__eq2,axiom,
    ! [R: set_Pr8226792695654363348_a_b_a,S: set_Pr8226792695654363348_a_b_a] :
      ( ( inf_in4702073847719567777_b_a_o
        @ ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ R )
        @ ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ ( inf_in8743711151036114626_a_b_a @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1208_inf__Int__eq2,axiom,
    ! [R: set_Pr2043227878544749668_a_b_a,S: set_Pr2043227878544749668_a_b_a] :
      ( ( inf_in8675142699724641169_b_a_o
        @ ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ R )
        @ ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ ( inf_in2539093205184043858_a_b_a @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1209_inf__Int__eq2,axiom,
    ! [R: set_Pr2084289749507318346_a_b_a,S: set_Pr2084289749507318346_a_b_a] :
      ( ( inf_in8672167748486136609_b_a_o
        @ ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ R )
        @ ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ S ) )
      = ( ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ ( inf_in1228299729535579356_a_b_a @ R @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1210_INF__Int__eq2,axiom,
    ! [S: set_se3561969970571257112_d_nat] :
      ( ( comple2971473640637493296_nat_o
        @ ( image_8176980452443277408_nat_o
          @ ^ [I: set_Pr8583615357880026082_d_nat,X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ I )
          @ S ) )
      = ( ^ [X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ ( comple4086198579289044507_d_nat @ S ) ) ) ) ).

% INF_Int_eq2
thf(fact_1211_INF__Int__eq2,axiom,
    ! [S: set_se3214616458824132456od_a_b] :
      ( ( comple2085224203390393140_a_b_o
        @ ( image_8629710542337338282_a_b_o
          @ ^ [I: set_Product_prod_a_b,X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ I )
          @ S ) )
      = ( ^ [X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ ( comple5126089992128148111od_a_b @ S ) ) ) ) ).

% INF_Int_eq2
thf(fact_1212_INF__Int__eq2,axiom,
    ! [S: set_se1240140977111469876_a_b_a] :
      ( ( comple5146229428000914234_b_a_o
        @ ( image_1759233996081757668_b_a_o
          @ ^ [I: set_Pr8226792695654363348_a_b_a,X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ I )
          @ S ) )
      = ( ^ [X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ ( comple3890089605970274907_a_b_a @ S ) ) ) ) ).

% INF_Int_eq2
thf(fact_1213_INF__Int__eq2,axiom,
    ! [S: set_se6657180468042755140_a_b_a] :
      ( ( comple4075320519695027370_b_a_o
        @ ( image_8355691765331406788_b_a_o
          @ ^ [I: set_Pr2043227878544749668_a_b_a,X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ I )
          @ S ) )
      = ( ^ [X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ ( comple1785656685207633515_a_b_a @ S ) ) ) ) ).

% INF_Int_eq2
thf(fact_1214_INF__Int__eq2,axiom,
    ! [S: set_se7745462473697426688_a_b_a] :
      ( ( comple664017283855950152_b_a_o
        @ ( image_741601757749928672_b_a_o
          @ ^ [I: set_Pr2084289749507318346_a_b_a,X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ I )
          @ S ) )
      = ( ^ [X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ ( comple280159177104926979_a_b_a @ S ) ) ) ) ).

% INF_Int_eq2
thf(fact_1215_Pair__vimage__Sigma,axiom,
    ! [X: a,A4: set_a,F: a > set_Pr7704721104634554677_d_nat] :
      ( ( ( member_a @ X @ A4 )
       => ( ( vimage1199531099893768680_d_nat @ ( produc5865182755557626534_d_nat @ X ) @ ( produc6456947282657855029_d_nat @ A4 @ F ) )
          = ( F @ X ) ) )
      & ( ~ ( member_a @ X @ A4 )
       => ( ( vimage1199531099893768680_d_nat @ ( produc5865182755557626534_d_nat @ X ) @ ( produc6456947282657855029_d_nat @ A4 @ F ) )
          = bot_bo6373193931821895113_d_nat ) ) ) ).

% Pair_vimage_Sigma
thf(fact_1216_Pair__vimage__Sigma,axiom,
    ! [X: a,A4: set_a,F: a > set_b] :
      ( ( ( member_a @ X @ A4 )
       => ( ( vimage4504484899009881380od_a_b @ ( product_Pair_a_b @ X ) @ ( product_Sigma_a_b @ A4 @ F ) )
          = ( F @ X ) ) )
      & ( ~ ( member_a @ X @ A4 )
       => ( ( vimage4504484899009881380od_a_b @ ( product_Pair_a_b @ X ) @ ( product_Sigma_a_b @ A4 @ F ) )
          = bot_bot_set_b ) ) ) ).

% Pair_vimage_Sigma
thf(fact_1217_Pair__vimage__Sigma,axiom,
    ! [X: produc1990351791824763820_d_nat,A4: set_Pr8583615357880026082_d_nat,F: produc1990351791824763820_d_nat > set_ma506406390364637853_a_b_a] :
      ( ( ( member6046481508403577027_d_nat @ X @ A4 )
       => ( ( vimage4486498852325311824_a_b_a @ ( produc1681437058057156268_a_b_a @ X ) @ ( produc3232134635143574877_a_b_a @ A4 @ F ) )
          = ( F @ X ) ) )
      & ( ~ ( member6046481508403577027_d_nat @ X @ A4 )
       => ( ( vimage4486498852325311824_a_b_a @ ( produc1681437058057156268_a_b_a @ X ) @ ( produc3232134635143574877_a_b_a @ A4 @ F ) )
          = bot_bo3815340846421858609_a_b_a ) ) ) ).

% Pair_vimage_Sigma
thf(fact_1218_Pair__vimage__Sigma,axiom,
    ! [X: list_P6684558232516339260_d_nat,A4: set_li2034412094485292274_d_nat,F: list_P6684558232516339260_d_nat > set_ma506406390364637853_a_b_a] :
      ( ( ( member136131075555046355_d_nat @ X @ A4 )
       => ( ( vimage6027918953727758688_a_b_a @ ( produc2616710729536263484_a_b_a @ X ) @ ( produc534304670606313325_a_b_a @ A4 @ F ) )
          = ( F @ X ) ) )
      & ( ~ ( member136131075555046355_d_nat @ X @ A4 )
       => ( ( vimage6027918953727758688_a_b_a @ ( produc2616710729536263484_a_b_a @ X ) @ ( produc534304670606313325_a_b_a @ A4 @ F ) )
          = bot_bo3815340846421858609_a_b_a ) ) ) ).

% Pair_vimage_Sigma
thf(fact_1219_Pair__vimage__Sigma,axiom,
    ! [X: a,A4: set_a,F: a > set_ma506406390364637853_a_b_a] :
      ( ( ( member_a @ X @ A4 )
       => ( ( vimage6771639459236215480_a_b_a @ ( produc1583556749610718734_a_b_a @ X ) @ ( produc6940724223187229_a_b_a @ A4 @ F ) )
          = ( F @ X ) ) )
      & ( ~ ( member_a @ X @ A4 )
       => ( ( vimage6771639459236215480_a_b_a @ ( produc1583556749610718734_a_b_a @ X ) @ ( produc6940724223187229_a_b_a @ A4 @ F ) )
          = bot_bo3815340846421858609_a_b_a ) ) ) ).

% Pair_vimage_Sigma
thf(fact_1220_Inf__INT__eq2,axiom,
    ( comple2971473640637493296_nat_o
    = ( ^ [S11: set_a_7514012873179435181_nat_o,X4: a,Y4: produc5456268773499131775_d_nat] : ( member6046481508403577027_d_nat @ ( produc5865182755557626534_d_nat @ X4 @ Y4 ) @ ( comple4086198579289044507_d_nat @ ( image_3130169668052712384_d_nat @ collec3895662986305461505_d_nat @ ( image_5276271149719299749_nat_o @ produc1905287380190322693_nat_o @ S11 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_1221_Inf__INT__eq2,axiom,
    ( comple2085224203390393140_a_b_o
    = ( ^ [S11: set_a_b_o,X4: a,Y4: b] : ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y4 ) @ ( comple5126089992128148111od_a_b @ ( image_5056655251382583084od_a_b @ collec3336397801687681299od_a_b @ ( image_5956505685088102973_a_b_o @ produc3537405659489547051_a_b_o @ S11 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_1222_Inf__INT__eq2,axiom,
    ( comple5146229428000914234_b_a_o
    = ( ^ [S11: set_Pr9119034101783098515_b_a_o,X4: produc1990351791824763820_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member4296969940938992029_a_b_a @ ( produc1681437058057156268_a_b_a @ X4 @ Y4 ) @ ( comple3890089605970274907_a_b_a @ ( image_1611519598313298116_a_b_a @ collec4442262398806403551_a_b_a @ ( image_5007618793744694903_b_a_o @ produc6407700812349938327_b_a_o @ S11 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_1223_Inf__INT__eq2,axiom,
    ( comple4075320519695027370_b_a_o
    = ( ^ [S11: set_li6280422973213826819_b_a_o,X4: list_P6684558232516339260_d_nat,Y4: mappin2365780136655028071_a_b_a] : ( member50691962224995117_a_b_a @ ( produc2616710729536263484_a_b_a @ X4 @ Y4 ) @ ( comple1785656685207633515_a_b_a @ ( image_4387920688708098276_a_b_a @ collec2527315220090610287_a_b_a @ ( image_2537138341275166167_b_a_o @ produc6164275942970966535_b_a_o @ S11 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_1224_Inf__INT__eq2,axiom,
    ( comple664017283855950152_b_a_o
    = ( ^ [S11: set_a_7235201166916292293_b_a_o,X4: a,Y4: mappin2365780136655028071_a_b_a] : ( member3694737009695996203_a_b_a @ ( produc1583556749610718734_a_b_a @ X4 @ Y4 ) @ ( comple280159177104926979_a_b_a @ ( image_4033343064979391424_a_b_a @ collec8037893338069451881_a_b_a @ ( image_3866804513976791973_b_a_o @ produc2053338206788791453_b_a_o @ S11 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_1225_inj__on__convol__ident,axiom,
    ! [F: a > produc5456268773499131775_d_nat,X8: set_a] :
      ( inj_on4929973081579242105_d_nat
      @ ^ [X4: a] : ( produc5865182755557626534_d_nat @ X4 @ ( F @ X4 ) )
      @ X8 ) ).

% inj_on_convol_ident
thf(fact_1226_inj__on__convol__ident,axiom,
    ! [F: a > b,X8: set_a] :
      ( inj_on8941660087544810907od_a_b
      @ ^ [X4: a] : ( product_Pair_a_b @ X4 @ ( F @ X4 ) )
      @ X8 ) ).

% inj_on_convol_ident
thf(fact_1227_inj__on__convol__ident,axiom,
    ! [F: produc1990351791824763820_d_nat > mappin2365780136655028071_a_b_a,X8: set_Pr8583615357880026082_d_nat] :
      ( inj_on1743855064036557333_a_b_a
      @ ^ [X4: produc1990351791824763820_d_nat] : ( produc1681437058057156268_a_b_a @ X4 @ ( F @ X4 ) )
      @ X8 ) ).

% inj_on_convol_ident
thf(fact_1228_inj__on__convol__ident,axiom,
    ! [F: list_P6684558232516339260_d_nat > mappin2365780136655028071_a_b_a,X8: set_li2034412094485292274_d_nat] :
      ( inj_on2443305404503267861_a_b_a
      @ ^ [X4: list_P6684558232516339260_d_nat] : ( produc2616710729536263484_a_b_a @ X4 @ ( F @ X4 ) )
      @ X8 ) ).

% inj_on_convol_ident
thf(fact_1229_inj__on__convol__ident,axiom,
    ! [F: a > mappin2365780136655028071_a_b_a,X8: set_a] :
      ( inj_on4314983988112238049_a_b_a
      @ ^ [X4: a] : ( produc1583556749610718734_a_b_a @ X4 @ ( F @ X4 ) )
      @ X8 ) ).

% inj_on_convol_ident
thf(fact_1230_swap__inj__on,axiom,
    ! [A4: set_Pr4625767069131929962_nat_a] :
      ( inj_on8219103925108403155_d_nat
      @ ( produc5894518052405473087_d_nat
        @ ^ [I: produc5456268773499131775_d_nat,J: a] : ( produc5865182755557626534_d_nat @ J @ I ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_1231_swap__inj__on,axiom,
    ! [A4: set_Product_prod_b_a] :
      ( inj_on6506093261728657715od_a_b
      @ ( produc4348216232050026237od_a_b
        @ ^ [I: b,J: a] : ( product_Pair_a_b @ J @ I ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_1232_swap__inj__on,axiom,
    ! [A4: set_Pr1266358368572761940_d_nat] :
      ( inj_on105552635532692977_a_b_a
      @ ( produc3170874359442458355_a_b_a
        @ ^ [I: mappin2365780136655028071_a_b_a,J: produc1990351791824763820_d_nat] : ( produc1681437058057156268_a_b_a @ J @ I ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_1233_swap__inj__on,axiom,
    ! [A4: set_Pr8192599956907258084_d_nat] :
      ( inj_on2672647652192256113_a_b_a
      @ ( produc5658256351460266227_a_b_a
        @ ^ [I: mappin2365780136655028071_a_b_a,J: list_P6684558232516339260_d_nat] : ( produc2616710729536263484_a_b_a @ J @ I ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_1234_swap__inj__on,axiom,
    ! [A4: set_Pr4349924792822425730_b_a_a] :
      ( inj_on7373616982336107219_a_b_a
      @ ( produc7265838118961918399_a_b_a
        @ ^ [I: mappin2365780136655028071_a_b_a,J: a] : ( produc1583556749610718734_a_b_a @ J @ I ) )
      @ A4 ) ).

% swap_inj_on
thf(fact_1235_apsnd__conv,axiom,
    ! [F: b > b,X: a,Y5: b] :
      ( ( product_apsnd_b_b_a @ F @ ( product_Pair_a_b @ X @ Y5 ) )
      = ( product_Pair_a_b @ X @ ( F @ Y5 ) ) ) ).

% apsnd_conv
thf(fact_1236_apsnd__conv,axiom,
    ! [F: b > mappin2365780136655028071_a_b_a,X: a,Y5: b] :
      ( ( produc7462416120108760596_b_a_a @ F @ ( product_Pair_a_b @ X @ Y5 ) )
      = ( produc1583556749610718734_a_b_a @ X @ ( F @ Y5 ) ) ) ).

% apsnd_conv
thf(fact_1237_apsnd__conv,axiom,
    ! [F: mappin2365780136655028071_a_b_a > b,X: a,Y5: mappin2365780136655028071_a_b_a] :
      ( ( produc5385714642348984356_a_b_a @ F @ ( produc1583556749610718734_a_b_a @ X @ Y5 ) )
      = ( product_Pair_a_b @ X @ ( F @ Y5 ) ) ) ).

% apsnd_conv
thf(fact_1238_apsnd__conv,axiom,
    ! [F: produc5456268773499131775_d_nat > b,X: a,Y5: produc5456268773499131775_d_nat] :
      ( ( produc6018477202091853116at_b_a @ F @ ( produc5865182755557626534_d_nat @ X @ Y5 ) )
      = ( product_Pair_a_b @ X @ ( F @ Y5 ) ) ) ).

% apsnd_conv
thf(fact_1239_apsnd__conv,axiom,
    ! [F: b > produc5456268773499131775_d_nat,X: a,Y5: b] :
      ( ( produc9039840829191257340_nat_a @ F @ ( product_Pair_a_b @ X @ Y5 ) )
      = ( produc5865182755557626534_d_nat @ X @ ( F @ Y5 ) ) ) ).

% apsnd_conv
thf(fact_1240_apsnd__conv,axiom,
    ! [F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,X: a,Y5: mappin2365780136655028071_a_b_a] :
      ( ( produc964556241393567920_b_a_a @ F @ ( produc1583556749610718734_a_b_a @ X @ Y5 ) )
      = ( produc1583556749610718734_a_b_a @ X @ ( F @ Y5 ) ) ) ).

% apsnd_conv
thf(fact_1241_apsnd__conv,axiom,
    ! [F: produc5456268773499131775_d_nat > mappin2365780136655028071_a_b_a,X: a,Y5: produc5456268773499131775_d_nat] :
      ( ( produc8248651739278155160_b_a_a @ F @ ( produc5865182755557626534_d_nat @ X @ Y5 ) )
      = ( produc1583556749610718734_a_b_a @ X @ ( F @ Y5 ) ) ) ).

% apsnd_conv
thf(fact_1242_apsnd__conv,axiom,
    ! [F: mappin2365780136655028071_a_b_a > produc5456268773499131775_d_nat,X: a,Y5: mappin2365780136655028071_a_b_a] :
      ( ( produc5693764782446549400_nat_a @ F @ ( produc1583556749610718734_a_b_a @ X @ Y5 ) )
      = ( produc5865182755557626534_d_nat @ X @ ( F @ Y5 ) ) ) ).

% apsnd_conv
thf(fact_1243_apsnd__conv,axiom,
    ! [F: produc5456268773499131775_d_nat > produc5456268773499131775_d_nat,X: a,Y5: produc5456268773499131775_d_nat] :
      ( ( produc351369503313934976_nat_a @ F @ ( produc5865182755557626534_d_nat @ X @ Y5 ) )
      = ( produc5865182755557626534_d_nat @ X @ ( F @ Y5 ) ) ) ).

% apsnd_conv
thf(fact_1244_apsnd__conv,axiom,
    ! [F: mappin2365780136655028071_a_b_a > mappin2365780136655028071_a_b_a,X: produc1990351791824763820_d_nat,Y5: mappin2365780136655028071_a_b_a] :
      ( ( produc8699478479990220410_d_nat @ F @ ( produc1681437058057156268_a_b_a @ X @ Y5 ) )
      = ( produc1681437058057156268_a_b_a @ X @ ( F @ Y5 ) ) ) ).

% apsnd_conv
thf(fact_1245_apfst__conv,axiom,
    ! [F: a > a,X: a,Y5: b] :
      ( ( product_apfst_a_a_b @ F @ ( product_Pair_a_b @ X @ Y5 ) )
      = ( product_Pair_a_b @ ( F @ X ) @ Y5 ) ) ).

% apfst_conv
thf(fact_1246_apfst__conv,axiom,
    ! [F: a > a,X: a,Y5: mappin2365780136655028071_a_b_a] :
      ( ( produc4084794728848183459_a_b_a @ F @ ( produc1583556749610718734_a_b_a @ X @ Y5 ) )
      = ( produc1583556749610718734_a_b_a @ ( F @ X ) @ Y5 ) ) ).

% apfst_conv
thf(fact_1247_apfst__conv,axiom,
    ! [F: a > a,X: a,Y5: produc5456268773499131775_d_nat] :
      ( ( produc5050640338377042619_d_nat @ F @ ( produc5865182755557626534_d_nat @ X @ Y5 ) )
      = ( produc5865182755557626534_d_nat @ ( F @ X ) @ Y5 ) ) ).

% apfst_conv
thf(fact_1248_apfst__conv,axiom,
    ! [F: produc1990351791824763820_d_nat > a,X: produc1990351791824763820_d_nat,Y5: mappin2365780136655028071_a_b_a] :
      ( ( produc3276499839099775357_a_b_a @ F @ ( produc1681437058057156268_a_b_a @ X @ Y5 ) )
      = ( produc1583556749610718734_a_b_a @ ( F @ X ) @ Y5 ) ) ).

% apfst_conv
thf(fact_1249_apfst__conv,axiom,
    ! [F: a > produc1990351791824763820_d_nat,X: a,Y5: mappin2365780136655028071_a_b_a] :
      ( ( produc6413990508196178775_a_b_a @ F @ ( produc1583556749610718734_a_b_a @ X @ Y5 ) )
      = ( produc1681437058057156268_a_b_a @ ( F @ X ) @ Y5 ) ) ).

% apfst_conv
thf(fact_1250_apfst__conv,axiom,
    ! [F: list_P6684558232516339260_d_nat > a,X: list_P6684558232516339260_d_nat,Y5: mappin2365780136655028071_a_b_a] :
      ( ( produc6859189103102910701_a_b_a @ F @ ( produc2616710729536263484_a_b_a @ X @ Y5 ) )
      = ( produc1583556749610718734_a_b_a @ ( F @ X ) @ Y5 ) ) ).

% apfst_conv
thf(fact_1251_apfst__conv,axiom,
    ! [F: a > list_P6684558232516339260_d_nat,X: a,Y5: mappin2365780136655028071_a_b_a] :
      ( ( produc4928477468678715495_a_b_a @ F @ ( produc1583556749610718734_a_b_a @ X @ Y5 ) )
      = ( produc2616710729536263484_a_b_a @ ( F @ X ) @ Y5 ) ) ).

% apfst_conv
thf(fact_1252_apfst__conv,axiom,
    ! [F: produc1990351791824763820_d_nat > produc1990351791824763820_d_nat,X: produc1990351791824763820_d_nat,Y5: mappin2365780136655028071_a_b_a] :
      ( ( produc6585218156587187453_a_b_a @ F @ ( produc1681437058057156268_a_b_a @ X @ Y5 ) )
      = ( produc1681437058057156268_a_b_a @ ( F @ X ) @ Y5 ) ) ).

% apfst_conv
thf(fact_1253_apfst__conv,axiom,
    ! [F: produc1990351791824763820_d_nat > list_P6684558232516339260_d_nat,X: produc1990351791824763820_d_nat,Y5: mappin2365780136655028071_a_b_a] :
      ( ( produc3293813856439549709_a_b_a @ F @ ( produc1681437058057156268_a_b_a @ X @ Y5 ) )
      = ( produc2616710729536263484_a_b_a @ ( F @ X ) @ Y5 ) ) ).

% apfst_conv
thf(fact_1254_apfst__conv,axiom,
    ! [F: list_P6684558232516339260_d_nat > produc1990351791824763820_d_nat,X: list_P6684558232516339260_d_nat,Y5: mappin2365780136655028071_a_b_a] :
      ( ( produc7361565777321820557_a_b_a @ F @ ( produc2616710729536263484_a_b_a @ X @ Y5 ) )
      = ( produc1681437058057156268_a_b_a @ ( F @ X ) @ Y5 ) ) ).

% apfst_conv
thf(fact_1255_apfst__convE,axiom,
    ! [Q: product_prod_a_b,F: a > a,P2: product_prod_a_b] :
      ( ( Q
        = ( product_apfst_a_a_b @ F @ P2 ) )
     => ~ ! [X3: a,Y3: b] :
            ( ( P2
              = ( product_Pair_a_b @ X3 @ Y3 ) )
           => ( Q
             != ( product_Pair_a_b @ ( F @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_1256_apfst__convE,axiom,
    ! [Q: produc7822699449245016212_a_b_a,F: a > a,P2: produc7822699449245016212_a_b_a] :
      ( ( Q
        = ( produc4084794728848183459_a_b_a @ F @ P2 ) )
     => ~ ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
            ( ( P2
              = ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) )
           => ( Q
             != ( produc1583556749610718734_a_b_a @ ( F @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_1257_apfst__convE,axiom,
    ! [Q: produc1990351791824763820_d_nat,F: a > a,P2: produc1990351791824763820_d_nat] :
      ( ( Q
        = ( produc5050640338377042619_d_nat @ F @ P2 ) )
     => ~ ! [X3: a,Y3: produc5456268773499131775_d_nat] :
            ( ( P2
              = ( produc5865182755557626534_d_nat @ X3 @ Y3 ) )
           => ( Q
             != ( produc5865182755557626534_d_nat @ ( F @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_1258_apfst__convE,axiom,
    ! [Q: produc7822699449245016212_a_b_a,F: produc1990351791824763820_d_nat > a,P2: produc6343861673719254644_a_b_a] :
      ( ( Q
        = ( produc3276499839099775357_a_b_a @ F @ P2 ) )
     => ~ ! [X3: produc1990351791824763820_d_nat,Y3: mappin2365780136655028071_a_b_a] :
            ( ( P2
              = ( produc1681437058057156268_a_b_a @ X3 @ Y3 ) )
           => ( Q
             != ( produc1583556749610718734_a_b_a @ ( F @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_1259_apfst__convE,axiom,
    ! [Q: produc6343861673719254644_a_b_a,F: a > produc1990351791824763820_d_nat,P2: produc7822699449245016212_a_b_a] :
      ( ( Q
        = ( produc6413990508196178775_a_b_a @ F @ P2 ) )
     => ~ ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
            ( ( P2
              = ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) )
           => ( Q
             != ( produc1681437058057156268_a_b_a @ ( F @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_1260_apfst__convE,axiom,
    ! [Q: produc7822699449245016212_a_b_a,F: list_P6684558232516339260_d_nat > a,P2: produc2062766750192416900_a_b_a] :
      ( ( Q
        = ( produc6859189103102910701_a_b_a @ F @ P2 ) )
     => ~ ! [X3: list_P6684558232516339260_d_nat,Y3: mappin2365780136655028071_a_b_a] :
            ( ( P2
              = ( produc2616710729536263484_a_b_a @ X3 @ Y3 ) )
           => ( Q
             != ( produc1583556749610718734_a_b_a @ ( F @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_1261_apfst__convE,axiom,
    ! [Q: produc2062766750192416900_a_b_a,F: a > list_P6684558232516339260_d_nat,P2: produc7822699449245016212_a_b_a] :
      ( ( Q
        = ( produc4928477468678715495_a_b_a @ F @ P2 ) )
     => ~ ! [X3: a,Y3: mappin2365780136655028071_a_b_a] :
            ( ( P2
              = ( produc1583556749610718734_a_b_a @ X3 @ Y3 ) )
           => ( Q
             != ( produc2616710729536263484_a_b_a @ ( F @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_1262_apfst__convE,axiom,
    ! [Q: produc6343861673719254644_a_b_a,F: produc1990351791824763820_d_nat > produc1990351791824763820_d_nat,P2: produc6343861673719254644_a_b_a] :
      ( ( Q
        = ( produc6585218156587187453_a_b_a @ F @ P2 ) )
     => ~ ! [X3: produc1990351791824763820_d_nat,Y3: mappin2365780136655028071_a_b_a] :
            ( ( P2
              = ( produc1681437058057156268_a_b_a @ X3 @ Y3 ) )
           => ( Q
             != ( produc1681437058057156268_a_b_a @ ( F @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_1263_apfst__convE,axiom,
    ! [Q: produc2062766750192416900_a_b_a,F: produc1990351791824763820_d_nat > list_P6684558232516339260_d_nat,P2: produc6343861673719254644_a_b_a] :
      ( ( Q
        = ( produc3293813856439549709_a_b_a @ F @ P2 ) )
     => ~ ! [X3: produc1990351791824763820_d_nat,Y3: mappin2365780136655028071_a_b_a] :
            ( ( P2
              = ( produc1681437058057156268_a_b_a @ X3 @ Y3 ) )
           => ( Q
             != ( produc2616710729536263484_a_b_a @ ( F @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE
thf(fact_1264_apfst__convE,axiom,
    ! [Q: produc6343861673719254644_a_b_a,F: list_P6684558232516339260_d_nat > produc1990351791824763820_d_nat,P2: produc2062766750192416900_a_b_a] :
      ( ( Q
        = ( produc7361565777321820557_a_b_a @ F @ P2 ) )
     => ~ ! [X3: list_P6684558232516339260_d_nat,Y3: mappin2365780136655028071_a_b_a] :
            ( ( P2
              = ( produc2616710729536263484_a_b_a @ X3 @ Y3 ) )
           => ( Q
             != ( produc1681437058057156268_a_b_a @ ( F @ X3 ) @ Y3 ) ) ) ) ).

% apfst_convE

% Helper facts (35)
thf(help_If_2_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
    ! [X: option_a,Y5: option_a] :
      ( ( if_option_a @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_Itf__a_J_T,axiom,
    ! [X: option_a,Y5: option_a] :
      ( ( if_option_a @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_T,axiom,
    ! [X: product_prod_a_b,Y5: product_prod_a_b] :
      ( ( if_Product_prod_a_b @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_T,axiom,
    ! [X: product_prod_a_b,Y5: product_prod_a_b] :
      ( ( if_Product_prod_a_b @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
    ! [X: list_P1396940483166286381od_a_a,Y5: list_P1396940483166286381od_a_a] :
      ( ( if_lis931442767461590515od_a_a @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_T,axiom,
    ! [X: list_P1396940483166286381od_a_a,Y5: list_P1396940483166286381od_a_a] :
      ( ( if_lis931442767461590515od_a_a @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_T,axiom,
    ! [X: list_P1467974523212632366od_a_b,Y5: list_P1467974523212632366od_a_b] :
      ( ( if_lis1002476807507936500od_a_b @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_T,axiom,
    ! [X: list_P1467974523212632366od_a_b,Y5: list_P1467974523212632366od_a_b] :
      ( ( if_lis1002476807507936500od_a_b @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: option3893003681250311888_d_nat,Y5: option3893003681250311888_d_nat] :
      ( ( if_opt5027858148196338570_d_nat @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: option3893003681250311888_d_nat,Y5: option3893003681250311888_d_nat] :
      ( ( if_opt5027858148196338570_d_nat @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_T,axiom,
    ! [X: produc7822699449245016212_a_b_a,Y5: produc7822699449245016212_a_b_a] :
      ( ( if_Pro7552986933024089678_a_b_a @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_T,axiom,
    ! [X: produc7822699449245016212_a_b_a,Y5: produc7822699449245016212_a_b_a] :
      ( ( if_Pro7552986933024089678_a_b_a @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_J_J_T,axiom,
    ! [X: list_P5700792096213675741od_a_b,Y5: list_P5700792096213675741od_a_b] :
      ( ( if_lis8754265742650888611od_a_b @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_J_J_T,axiom,
    ! [X: list_P5700792096213675741od_a_b,Y5: list_P5700792096213675741od_a_b] :
      ( ( if_lis8754265742650888611od_a_b @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_J_T,axiom,
    ! [X: list_P2992919015626629540_a_b_a,Y5: list_P2992919015626629540_a_b_a] :
      ( ( if_lis7797837208463966814_a_b_a @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_J_T,axiom,
    ! [X: list_P2992919015626629540_a_b_a,Y5: list_P2992919015626629540_a_b_a] :
      ( ( if_lis7797837208463966814_a_b_a @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X: produc1990351791824763820_d_nat,Y5: produc1990351791824763820_d_nat] :
      ( ( if_Pro3152534550563304550_d_nat @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X: produc1990351791824763820_d_nat,Y5: produc1990351791824763820_d_nat] :
      ( ( if_Pro3152534550563304550_d_nat @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J_J_T,axiom,
    ! [X: list_P6684558232516339260_d_nat,Y5: list_P6684558232516339260_d_nat] :
      ( ( if_lis7731207453108189686_d_nat @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J_J_T,axiom,
    ! [X: list_P6684558232516339260_d_nat,Y5: list_P6684558232516339260_d_nat] :
      ( ( if_lis7731207453108189686_d_nat @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_J_J_J_T,axiom,
    ! [X: list_P3172577687508813921_a_b_a,Y5: list_P3172577687508813921_a_b_a] :
      ( ( if_lis3166759124437449243_a_b_a @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_J_J_J_T,axiom,
    ! [X: list_P3172577687508813921_a_b_a,Y5: list_P3172577687508813921_a_b_a] :
      ( ( if_lis3166759124437449243_a_b_a @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_Mtf__a_J_J_J_T,axiom,
    ! [X: list_P3594502398639369124_b_a_a,Y5: list_P3594502398639369124_b_a_a] :
      ( ( if_lis2561630526582691434_b_a_a @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_Mtf__a_J_J_J_T,axiom,
    ! [X: list_P3594502398639369124_b_a_a,Y5: list_P3594502398639369124_b_a_a] :
      ( ( if_lis2561630526582691434_b_a_a @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J_J_J_J_T,axiom,
    ! [X: list_P9063253934932097401_d_nat,Y5: list_P9063253934932097401_d_nat] :
      ( ( if_lis3214374446479449651_d_nat @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J_J_J_J_T,axiom,
    ! [X: list_P9063253934932097401_d_nat,Y5: list_P9063253934932097401_d_nat] :
      ( ( if_lis3214374446479449651_d_nat @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_T,axiom,
    ! [X: produc6343861673719254644_a_b_a,Y5: produc6343861673719254644_a_b_a] :
      ( ( if_Pro8612401522923744058_a_b_a @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_T,axiom,
    ! [X: produc6343861673719254644_a_b_a,Y5: produc6343861673719254644_a_b_a] :
      ( ( if_Pro8612401522923744058_a_b_a @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_J_T,axiom,
    ! [X: list_P4793483941206178810_a_b_a,Y5: list_P4793483941206178810_a_b_a] :
      ( ( if_lis8426428089627639232_a_b_a @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_J_T,axiom,
    ! [X: list_P4793483941206178810_a_b_a,Y5: list_P4793483941206178810_a_b_a] :
      ( ( if_lis8426428089627639232_a_b_a @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J_J_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_T,axiom,
    ! [X: produc2062766750192416900_a_b_a,Y5: produc2062766750192416900_a_b_a] :
      ( ( if_Pro5695710898613877322_a_b_a @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J_J_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_T,axiom,
    ! [X: produc2062766750192416900_a_b_a,Y5: produc2062766750192416900_a_b_a] :
      ( ( if_Pro5695710898613877322_a_b_a @ $true @ X @ Y5 )
      = X ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J_J_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J_J_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_J_T,axiom,
    ! [X: list_P7611574810425128074_a_b_a,Y5: list_P7611574810425128074_a_b_a] :
      ( ( if_lis6818758761248955216_a_b_a @ $false @ X @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__c_Mt__Option__Ooption_It__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J_J_J_J_J_Mt__Mapping__Omapping_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_J_J_T,axiom,
    ! [X: list_P7611574810425128074_a_b_a,Y5: list_P7611574810425128074_a_b_a] :
      ( ( if_lis6818758761248955216_a_b_a @ $true @ X @ Y5 )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( s2
      = ( mmap_f8625263917883502961_d_nat @ s
        @ ( produc5272039204407661311_d_nat
          @ ^ [X4: a,V: produc5456268773499131775_d_nat] : ( produc5865182755557626534_d_nat @ ( step @ X4 @ bs ) @ ( drop_cur_c_d @ i @ V ) ) )
        @ ^ [X4: produc5456268773499131775_d_nat,Y4: produc5456268773499131775_d_nat] : X4
        @ r ) )
    & ! [Q3: a,Bs3: b] :
        ( case_option_o_a @ $true
        @ ( ^ [Y: a,Z: a] : ( Y = Z )
          @ ( step @ Q3 @ Bs3 ) )
        @ ( lookup863726980359885586_a_b_a @ st2 @ ( product_Pair_a_b @ Q3 @ Bs3 ) ) ) ) ).

%------------------------------------------------------------------------------