%------------------------------------------------------------------------------
% File     : ITP144_7 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 syntactic export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : iterate.ax [Gau19]
%          : HL4144_7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  661 ( 195 unt; 143 typ;   0 def)
%            Number of atoms       : 1833 ( 813 equ)
%            Maximal formula atoms :   28 (   2 avg)
%            Number of connectives : 1405 (  90   ~;   8   |; 439   &)
%                                         ( 125 <=>; 743  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   34 (   7 avg)
%            Maximal term depth    :   11 (   2 avg)
%            Number of types       :    4 (   3 usr)
%            Number of type conns  :  119 (  68   >;  51   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :  139 ( 139 usr;  36 con; 0-7 aty)
%            Number of variables   : 3247 (3092   !;  59   ?;3247   :)
%                                         (  96  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: $tType ).

tff(tyop_2Emin_2Efun,type,
    tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).

tff(tyop_2Enum_2Enum,type,
    tyop_2Enum_2Enum: $tType ).

tff(tyop_2Epair_2Eprod,type,
    tyop_2Epair_2Eprod: ( $tType * $tType ) > $tType ).

tff(tyop_2Erealax_2Ereal,type,
    tyop_2Erealax_2Ereal: $tType ).

tff(app_2E2,type,
    app_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).

tff(p,type,
    p: tyop_2Emin_2Ebool > $o ).

tff(combin_i_2E0,type,
    combin_i_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).

tff(combin_k_2E0,type,
    combin_k_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)) ).

tff(combin_s_2E0,type,
    combin_s_2E0: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))) ).

tff(c_2Ebool_2E_21_2E0,type,
    c_2Ebool_2E_21_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2E_21_2E1,type,
    c_2Ebool_2E_21_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).

tff(c_2Earithmetic_2E_2A_2E0,type,
    c_2Earithmetic_2E_2A_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2E_2A_2E2,type,
    c_2Earithmetic_2E_2A_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Earithmetic_2E_2B_2E0,type,
    c_2Earithmetic_2E_2B_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2E_2B_2E2,type,
    c_2Earithmetic_2E_2B_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Epair_2E_2C_2E0,type,
    c_2Epair_2E_2C_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))) ).

tff(c_2Epair_2E_2C_2E2,type,
    c_2Epair_2E_2C_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * A_27b ) > tyop_2Epair_2Eprod(A_27a,A_27b) ) ).

tff(c_2Earithmetic_2E_2D_2E0,type,
    c_2Earithmetic_2E_2D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2E_2D_2E2,type,
    c_2Earithmetic_2E_2D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Eiterate_2E_2E_2E_2E0,type,
    c_2Eiterate_2E_2E_2E_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))) ).

tff(c_2Eiterate_2E_2E_2E_2E2,type,
    c_2Eiterate_2E_2E_2E_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool) ).

tff(c_2Ereal_2E_2F_2E0,type,
    c_2Ereal_2E_2F_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).

tff(c_2Ereal_2E_2F_2E2,type,
    c_2Ereal_2E_2F_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).

tff(c_2Ebool_2E_2F_5C_2E0,type,
    c_2Ebool_2E_2F_5C_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).

tff(c_2Ebool_2E_2F_5C_2E2,type,
    c_2Ebool_2E_2F_5C_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).

tff(c_2Enum_2E0_2E0,type,
    c_2Enum_2E0_2E0: tyop_2Enum_2Enum ).

tff(c_2Eprim__rec_2E_3C_2E0,type,
    c_2Eprim__rec_2E_3C_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).

tff(c_2Eprim__rec_2E_3C_2E2,type,
    c_2Eprim__rec_2E_3C_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).

tff(c_2Earithmetic_2E_3C_3D_2E0,type,
    c_2Earithmetic_2E_3C_3D_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).

tff(c_2Earithmetic_2E_3C_3D_2E2,type,
    c_2Earithmetic_2E_3C_3D_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ).

tff(c_2Emin_2E_3D_2E0,type,
    c_2Emin_2E_3D_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).

tff(c_2Emin_2E_3D_2E2,type,
    c_2Emin_2E_3D_2E2: 
      !>[A_27a: $tType] : ( ( A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).

tff(c_2Emin_2E_3D_3D_3E_2E0,type,
    c_2Emin_2E_3D_3D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).

tff(c_2Emin_2E_3D_3D_3E_2E2,type,
    c_2Emin_2E_3D_3D_3E_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).

tff(c_2Ebool_2E_3F_2E0,type,
    c_2Ebool_2E_3F_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2E_3F_2E1,type,
    c_2Ebool_2E_3F_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).

tff(c_2Ebool_2E_3F_21_2E0,type,
    c_2Ebool_2E_3F_21_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2E_3F_21_2E1,type,
    c_2Ebool_2E_3F_21_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).

tff(c_2Emin_2E_40_2E0,type,
    c_2Emin_2E_40_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a) ).

tff(c_2Emin_2E_40_2E1,type,
    c_2Emin_2E_40_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > A_27a ) ).

tff(c_2Epred__set_2EBIGUNION_2E0,type,
    c_2Epred__set_2EBIGUNION_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).

tff(c_2Epred__set_2EBIGUNION_2E1,type,
    c_2Epred__set_2EBIGUNION_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).

tff(c_2Earithmetic_2EBIT1_2E0,type,
    c_2Earithmetic_2EBIT1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2EBIT1_2E1,type,
    c_2Earithmetic_2EBIT1_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Earithmetic_2EBIT2_2E0,type,
    c_2Earithmetic_2EBIT2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2EBIT2_2E1,type,
    c_2Earithmetic_2EBIT2_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Epred__set_2ECARD_2E0,type,
    c_2Epred__set_2ECARD_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Enum_2Enum) ).

tff(c_2Epred__set_2ECARD_2E1,type,
    c_2Epred__set_2ECARD_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Enum_2Enum ) ).

tff(c_2Ebool_2ECOND_2E0,type,
    c_2Ebool_2ECOND_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).

tff(c_2Ebool_2ECOND_2E3,type,
    c_2Ebool_2ECOND_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Ebool * A_27a * A_27a ) > A_27a ) ).

tff(c_2Epred__set_2EDELETE_2E0,type,
    c_2Epred__set_2EDELETE_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).

tff(c_2Epred__set_2EDELETE_2E2,type,
    c_2Epred__set_2EDELETE_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * A_27a ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).

tff(c_2Epred__set_2EDIFF_2E0,type,
    c_2Epred__set_2EDIFF_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).

tff(c_2Epred__set_2EDIFF_2E2,type,
    c_2Epred__set_2EDIFF_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).

tff(c_2Epred__set_2EDISJOINT_2E0,type,
    c_2Epred__set_2EDISJOINT_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).

tff(c_2Epred__set_2EDISJOINT_2E2,type,
    c_2Epred__set_2EDISJOINT_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).

tff(c_2Earithmetic_2EDIV_2E0,type,
    c_2Earithmetic_2EDIV_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2EDIV_2E2,type,
    c_2Earithmetic_2EDIV_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Epred__set_2EEMPTY_2E0,type,
    c_2Epred__set_2EEMPTY_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2EF_2E0,type,
    c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).

tff(c_2Epred__set_2EFINITE_2E0,type,
    c_2Epred__set_2EFINITE_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).

tff(c_2Epred__set_2EFINITE_2E1,type,
    c_2Epred__set_2EFINITE_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).

tff(c_2Eiterate_2EFINREC_2E0,type,
    c_2Eiterate_2EFINREC_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))))) ).

tff(c_2Eiterate_2EFINREC_2E5,type,
    c_2Eiterate_2EFINREC_2E5: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)) * A_27b * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * A_27b * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ) ).

tff(c_2Epred__set_2EGSPEC_2E0,type,
    c_2Epred__set_2EGSPEC_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).

tff(c_2Epred__set_2EGSPEC_2E1,type,
    c_2Epred__set_2EGSPEC_2E1: 
      !>[A_27a: $tType,A_27b: $tType] : ( tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).

tff(c_2Ecardinal_2EHAS__SIZE_2E0,type,
    c_2Ecardinal_2EHAS__SIZE_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).

tff(c_2Ecardinal_2EHAS__SIZE_2E2,type,
    c_2Ecardinal_2EHAS__SIZE_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ) ).

tff(c_2Epred__set_2EIMAGE_2E0,type,
    c_2Epred__set_2EIMAGE_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))) ).

tff(c_2Epred__set_2EIMAGE_2E2,type,
    c_2Epred__set_2EIMAGE_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool) ) ).

tff(c_2Ebool_2EIN_2E0,type,
    c_2Ebool_2EIN_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).

tff(c_2Ebool_2EIN_2E2,type,
    c_2Ebool_2EIN_2E2: 
      !>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).

tff(c_2Epred__set_2EINSERT_2E0,type,
    c_2Epred__set_2EINSERT_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).

tff(c_2Epred__set_2EINSERT_2E2,type,
    c_2Epred__set_2EINSERT_2E2: 
      !>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).

tff(c_2Epred__set_2EINTER_2E0,type,
    c_2Epred__set_2EINTER_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).

tff(c_2Epred__set_2EINTER_2E2,type,
    c_2Epred__set_2EINTER_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).

tff(c_2Eiterate_2EITSET_2E0,type,
    c_2Eiterate_2EITSET_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,A_27a))) ).

tff(c_2Eiterate_2EITSET_2E3,type,
    c_2Eiterate_2EITSET_2E3: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27a)) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool) * A_27a ) > A_27a ) ).

tff(c_2Earithmetic_2EMOD_2E0,type,
    c_2Earithmetic_2EMOD_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2EMOD_2E2,type,
    c_2Earithmetic_2EMOD_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Earithmetic_2ENUMERAL_2E0,type,
    c_2Earithmetic_2ENUMERAL_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2ENUMERAL_2E1,type,
    c_2Earithmetic_2ENUMERAL_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Epred__set_2ESUBSET_2E0,type,
    c_2Epred__set_2ESUBSET_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).

tff(c_2Epred__set_2ESUBSET_2E2,type,
    c_2Epred__set_2ESUBSET_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).

tff(c_2Enum_2ESUC_2E0,type,
    c_2Enum_2ESUC_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Enum_2ESUC_2E1,type,
    c_2Enum_2ESUC_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Eiterate_2ESum_2E0,type,
    c_2Eiterate_2ESum_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal)) ).

tff(c_2Eiterate_2ESum_2E2,type,
    c_2Eiterate_2ESum_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal) ) > tyop_2Erealax_2Ereal ) ).

tff(c_2Ebool_2ET_2E0,type,
    c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).

tff(c_2Epair_2EUNCURRY_2E0,type,
    c_2Epair_2EUNCURRY_2E0: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c)) ).

tff(c_2Epair_2EUNCURRY_2E1,type,
    c_2Epair_2EUNCURRY_2E1: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)) > tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c) ) ).

tff(c_2Epred__set_2EUNION_2E0,type,
    c_2Epred__set_2EUNION_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).

tff(c_2Epred__set_2EUNION_2E2,type,
    c_2Epred__set_2EUNION_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).

tff(c_2Epred__set_2EUNIV_2E0,type,
    c_2Epred__set_2EUNIV_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ).

tff(c_2Earithmetic_2EZERO_2E0,type,
    c_2Earithmetic_2EZERO_2E0: tyop_2Enum_2Enum ).

tff(c_2Ebool_2E_5C_2F_2E0,type,
    c_2Ebool_2E_5C_2F_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).

tff(c_2Ebool_2E_5C_2F_2E2,type,
    c_2Ebool_2E_5C_2F_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).

tff(c_2Ereal_2Eabs_2E0,type,
    c_2Ereal_2Eabs_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).

tff(c_2Ereal_2Eabs_2E1,type,
    c_2Ereal_2Eabs_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).

tff(c_2Eiterate_2Einf_2E0,type,
    c_2Eiterate_2Einf_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal) ).

tff(c_2Eiterate_2Einf_2E1,type,
    c_2Eiterate_2Einf_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Erealax_2Ereal ).

tff(c_2Eiterate_2Eiterate_2E0,type,
    c_2Eiterate_2Eiterate_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),A_27b))) ).

tff(c_2Eiterate_2Eiterate_2E1,type,
    c_2Eiterate_2Eiterate_2E1: 
      !>[A_27a: $tType,A_27b: $tType] : ( tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)) > tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),A_27b)) ) ).

tff(c_2Eiterate_2Eiterate_2E3,type,
    c_2Eiterate_2Eiterate_2E3: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,A_27b) ) > A_27b ) ).

tff(c_2Ereal_2Emax_2E0,type,
    c_2Ereal_2Emax_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).

tff(c_2Ereal_2Emax_2E2,type,
    c_2Ereal_2Emax_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).

tff(c_2Ereal_2Emin_2E0,type,
    c_2Ereal_2Emin_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).

tff(c_2Ereal_2Emin_2E2,type,
    c_2Ereal_2Emin_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).

tff(c_2Eiterate_2Emonoidal_2E0,type,
    c_2Eiterate_2Emonoidal_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Ebool) ).

tff(c_2Eiterate_2Emonoidal_2E1,type,
    c_2Eiterate_2Emonoidal_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)) > tyop_2Emin_2Ebool ) ).

tff(c_2Eiterate_2Eneutral_2E0,type,
    c_2Eiterate_2Eneutral_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),A_27a) ).

tff(c_2Eiterate_2Eneutral_2E1,type,
    c_2Eiterate_2Eneutral_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)) > A_27a ) ).

tff(c_2Eiterate_2Ensum_2E0,type,
    c_2Eiterate_2Ensum_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Enum_2Enum)) ).

tff(c_2Eiterate_2Ensum_2E2,type,
    c_2Eiterate_2Ensum_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum) ) > tyop_2Enum_2Enum ) ).

tff(c_2Ecombin_2Eo_2E0,type,
    c_2Ecombin_2Eo_2E0: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(A_27a,A_27b))) ).

tff(c_2Ecombin_2Eo_2E2,type,
    c_2Ecombin_2Eo_2E2: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Emin_2Efun(A_27c,A_27b) * tyop_2Emin_2Efun(A_27a,A_27c) ) > tyop_2Emin_2Efun(A_27a,A_27b) ) ).

tff(c_2Eiterate_2Epolynomial__function_2E0,type,
    c_2Eiterate_2Epolynomial__function_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool) ).

tff(c_2Eiterate_2Epolynomial__function_2E1,type,
    c_2Eiterate_2Epolynomial__function_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) > tyop_2Emin_2Ebool ).

tff(c_2Ereal_2Epow_2E0,type,
    c_2Ereal_2Epow_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)) ).

tff(c_2Ereal_2Epow_2E2,type,
    c_2Ereal_2Epow_2E2: ( tyop_2Erealax_2Ereal * tyop_2Enum_2Enum ) > tyop_2Erealax_2Ereal ).

tff(c_2Erealax_2Ereal__add_2E0,type,
    c_2Erealax_2Ereal__add_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).

tff(c_2Erealax_2Ereal__add_2E2,type,
    c_2Erealax_2Ereal__add_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).

tff(c_2Ereal_2Ereal__gt_2E0,type,
    c_2Ereal_2Ereal__gt_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).

tff(c_2Ereal_2Ereal__gt_2E2,type,
    c_2Ereal_2Ereal__gt_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).

tff(c_2Erealax_2Ereal__lt_2E0,type,
    c_2Erealax_2Ereal__lt_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).

tff(c_2Erealax_2Ereal__lt_2E2,type,
    c_2Erealax_2Ereal__lt_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).

tff(c_2Ereal_2Ereal__lte_2E0,type,
    c_2Ereal_2Ereal__lte_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).

tff(c_2Ereal_2Ereal__lte_2E2,type,
    c_2Ereal_2Ereal__lte_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).

tff(c_2Erealax_2Ereal__mul_2E0,type,
    c_2Erealax_2Ereal__mul_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).

tff(c_2Erealax_2Ereal__mul_2E2,type,
    c_2Erealax_2Ereal__mul_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).

tff(c_2Erealax_2Ereal__neg_2E0,type,
    c_2Erealax_2Ereal__neg_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).

tff(c_2Erealax_2Ereal__neg_2E1,type,
    c_2Erealax_2Ereal__neg_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).

tff(c_2Ereal_2Ereal__of__num_2E0,type,
    c_2Ereal_2Ereal__of__num_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) ).

tff(c_2Ereal_2Ereal__of__num_2E1,type,
    c_2Ereal_2Ereal__of__num_2E1: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).

tff(c_2Ereal_2Ereal__sub_2E0,type,
    c_2Ereal_2Ereal__sub_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).

tff(c_2Ereal_2Ereal__sub_2E2,type,
    c_2Ereal_2Ereal__sub_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).

tff(c_2Ereal_2Esup_2E0,type,
    c_2Ereal_2Esup_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal) ).

tff(c_2Ereal_2Esup_2E1,type,
    c_2Ereal_2Esup_2E1: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool) > tyop_2Erealax_2Ereal ).

tff(c_2Eiterate_2Esupport_2E0,type,
    c_2Eiterate_2Esupport_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))) ).

tff(c_2Eiterate_2Esupport_2E3,type,
    c_2Eiterate_2Esupport_2E3: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)) * tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).

tff(c_2Ebool_2E_7E_2E0,type,
    c_2Ebool_2E_7E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2E_7E_2E1,type,
    c_2Ebool_2E_7E_2E1: tyop_2Emin_2Ebool > tyop_2Emin_2Ebool ).

tff(thm_2Eextra_2Dho_2Eeq__ext,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
      ( ! [V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0) = app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0) )
     => ( V0f_2E0 = V1g_2E0 ) ) ).

tff(thm_2Eextra_2Dho_2Eboolext,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(V0_2E0)
      <=> p(V1_2E0) )
     => ( V0_2E0 = V1_2E0 ) ) ).

tff(thm_2Eextra_2Dho_2Etruth,axiom,
    p(c_2Ebool_2ET_2E0) ).

tff(thm_2Eextra_2Dho_2Enotfalse,axiom,
    ~ p(c_2Ebool_2EF_2E0) ).

tff(thm_2Eextra_2Dho_2Ebool__cases__ax,axiom,
    ! [V0t_2E0: tyop_2Emin_2Ebool] :
      ( ( V0t_2E0 = c_2Ebool_2ET_2E0 )
      | ( V0t_2E0 = c_2Ebool_2EF_2E0 ) ) ).

tff(thm_2Eextra_2Dho_2Ei__thm,axiom,
    ! [A_27a: $tType,V0x_2E0: A_27a] : ( app_2E2(A_27a,A_27a,combin_i_2E0(A_27a),V0x_2E0) = V0x_2E0 ) ).

tff(thm_2Eextra_2Dho_2Ek__thm,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27a),combin_k_2E0(A_27a,A_27b),V0x_2E0),V1y_2E0) = V0x_2E0 ) ).

tff(thm_2Eextra_2Dho_2Es__thm,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),combin_s_2E0(A_27a,A_27b,A_27c),V0f_2E0),V1g_2E0),V2x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V2x_2E0),app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0)) ) ).

tff(logicdef_2E_2F_5C,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Ebool_2E_2F_5C_2E2(V0_2E0,V1_2E0))
    <=> ( p(V0_2E0)
        & p(V1_2E0) ) ) ).

tff(logicdef_2E_5C_2F,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Ebool_2E_5C_2F_2E2(V0_2E0,V1_2E0))
    <=> ( p(V0_2E0)
        | p(V1_2E0) ) ) ).

tff(logicdef_2E_7E,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Ebool_2E_7E_2E1(V0_2E0))
    <=> ~ p(V0_2E0) ) ).

tff(logicdef_2E_3D_3D_3E,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Emin_2E_3D_3D_3E_2E2(V0_2E0,V1_2E0))
    <=> ( p(V0_2E0)
       => p(V1_2E0) ) ) ).

tff(logicdef_2E_3D,axiom,
    ! [A_27a: $tType,V0_2E0: A_27a,V1_2E0: A_27a] :
      ( p(c_2Emin_2E_3D_2E2(A_27a,V0_2E0,V1_2E0))
    <=> ( V0_2E0 = V1_2E0 ) ) ).

tff(quantdef_2E_21,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( p(c_2Ebool_2E_21_2E1(A_27a,V0f_2E0))
    <=> ! [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).

tff(quantdef_2E_3F,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( p(c_2Ebool_2E_3F_2E1(A_27a,V0f_2E0))
    <=> ? [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).

tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(A_27a),X0_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2E_2A_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2A_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2A_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2E_2B_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2B_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a,X1_2E0: A_27b] : ( c_2Epair_2E_2C_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Epair_2E_2C_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(A_27a,tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27b_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [A_27b: $tType,X0_2E0: A_27b,X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(A_27b,tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Epair_2E_2C_2E0(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2E_2D_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2D_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Eiterate_2E_2E_2E_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2E_2E_2E_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Eiterate_2E_2E_2E_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ereal_2E_2F_2E2,axiom,
    ! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2E_2F_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ereal_2E_2F_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(X0_2E0)
        & p(X1_2E0) )
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_2F_5C_2E0,X0_2E0),X1_2E0)) ) ).

tff(arityeq2_2Ec_2Eprim__rec_2E_3C_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Eprim__rec_2E_3C_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2E_3C_3D_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_3C_3D_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3C_3D_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: A_27a] :
      ( ( X0_2E0 = X1_2E0 )
    <=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(A_27a),X0_2E0),X1_2E0)) ) ).

tff(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(X0_2E0)
       => p(X1_2E0) )
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Emin_2E_3D_3D_3E_2E0,X0_2E0),X1_2E0)) ) ).

tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ebool_2E_3F_21_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_21_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a,c_2Emin_2E_40_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Efun_28A_27b_2Ctyop_2Emin_2Ebool_29_2CA_27a_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a),tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a),c_2Emin_2E_40_2E0(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a)),X0_2E0) ) ).

tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,c_2Emin_2E_40_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2EBIGUNION_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EBIGUNION_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Epred__set_2EBIGUNION_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2EBIT1_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT1_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2EBIT2_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT2_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT2_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2ECARD_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ECARD_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,c_2Epred__set_2ECARD_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2ECARD_2E1_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ECARD_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,c_2Epred__set_2ECARD_2E0(A_27b),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2ECARD_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ECARD_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,c_2Epred__set_2ECARD_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2ECARD_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,c_2Epred__set_2ECARD_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: A_27a,X2_2E0: A_27a] : ( c_2Ebool_2ECOND_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),c_2Ebool_2ECOND_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: A_27b,X2_2E0: A_27b] : ( c_2Ebool_2ECOND_2E3(A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),c_2Ebool_2ECOND_2E0(A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Emin_2Efun_28A_27b_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),c_2Ebool_2ECOND_2E0(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Emin_2Efun_28A_27c_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [A_27c: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),X2_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)] : ( c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),c_2Ebool_2ECOND_2E0(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Emin_2Efun_28tyop_2Enum_2Enum_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),c_2Ebool_2ECOND_2E0(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Enum_2Enum,X2_2E0: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Ebool_2ECOND_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Erealax_2Ereal,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Erealax_2Ereal] : ( c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ebool_2ECOND_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EDELETE_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: A_27a] : ( c_2Epred__set_2EDELETE_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EDELETE_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EDELETE_2E2_2Emono_2EA_27d,axiom,
    ! [A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),X1_2E0: A_27d] : ( c_2Epred__set_2EDELETE_2E2(A_27d,X0_2E0,X1_2E0) = app_2E2(A_27d,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),c_2Epred__set_2EDELETE_2E0(A_27d),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EDIFF_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDIFF_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EDIFF_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EDIFF_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDIFF_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),c_2Epred__set_2EDIFF_2E0(A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EDIFF_2E2_2Emono_2EA_27g,axiom,
    ! [A_27g: $tType,X0_2E0: tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDIFF_2E2(A_27g,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool)),c_2Epred__set_2EDIFF_2E0(A_27g),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EDISJOINT_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDISJOINT_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EDISJOINT_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EDISJOINT_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDISJOINT_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EDISJOINT_2E0(A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EDISJOINT_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EDISJOINT_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EDISJOINT_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2EDIV_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EDIV_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EDIV_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(A_27b),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).

tff(arityeq5_2Ec_2Eiterate_2EFINREC_2E5_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),X1_2E0: A_27b,X2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X3_2E0: A_27b,X4_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2EFINREC_2E5(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0,X3_2E0,X4_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))),c_2Eiterate_2EFINREC_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0),X3_2E0),X4_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2EA_27a_20mono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(A_27a,A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(A_27b,A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(A_27b,A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2EA_27b_20mono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(A_27b,A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(A_27b,A_27b),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2EA_27b_20mono_2Etyop_2Epair_2Eprod_28A_27c_2CA_27d_29,axiom,
    ! [A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27d),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(A_27b,tyop_2Epair_2Eprod(A_27c,A_27d),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27d),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(A_27b,tyop_2Epair_2Eprod(A_27c,A_27d)),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2EA_27b_20mono_2Etyop_2Epair_2Eprod_28A_27e_2Ctyop_2Epair_2Eprod_28A_27f_2CA_27g_29_29,axiom,
    ! [A_27b: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g)),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(A_27b,tyop_2Epair_2Eprod(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g)),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g)),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(A_27b,tyop_2Epair_2Eprod(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g))),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0) ) ).

tff(arityeq2_2Ec_2Ecardinal_2EHAS__SIZE_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Enum_2Enum] : ( c_2Ecardinal_2EHAS__SIZE_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Ecardinal_2EHAS__SIZE_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ecardinal_2EHAS__SIZE_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Enum_2Enum] : ( c_2Ecardinal_2EHAS__SIZE_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Ecardinal_2EHAS__SIZE_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2EA_27h_20mono_2EA_27i,axiom,
    ! [A_27h: $tType,A_27i: $tType,X0_2E0: tyop_2Emin_2Efun(A_27h,A_27i),X1_2E0: tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(A_27h,A_27i,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27h,A_27i),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(A_27h,A_27i),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(tyop_2Enum_2Enum,A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Epred__set_2EIMAGE_2E0(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: A_27b,X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
    ! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EINSERT_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINSERT_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EINSERT_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EINSERT_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: A_27b,X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINSERT_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),c_2Epred__set_2EINSERT_2E0(A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EINSERT_2E2_2Emono_2EA_27c,axiom,
    ! [A_27c: $tType,X0_2E0: A_27c,X1_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINSERT_2E2(A_27c,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),c_2Epred__set_2EINSERT_2E0(A_27c),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EINSERT_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINSERT_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Epred__set_2EINSERT_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EINSERT_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
    ! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Epred__set_2EINSERT_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EINTER_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINTER_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EINTER_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EINTER_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINTER_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),c_2Epred__set_2EINTER_2E0(A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EINTER_2E2_2Emono_2EA_27f,axiom,
    ! [A_27f: $tType,X0_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EINTER_2E2(A_27f,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool)),c_2Epred__set_2EINTER_2E0(A_27f),X0_2E0),X1_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2EITSET_2E3_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X2_2E0: A_27b] : ( c_2Eiterate_2EITSET_2E3(A_27b,A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27b,A_27b,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,A_27b),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,A_27b)),c_2Eiterate_2EITSET_2E0(A_27b,A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2EITSET_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27a)),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X2_2E0: A_27a] : ( c_2Eiterate_2EITSET_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,A_27a)),c_2Eiterate_2EITSET_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2EMOD_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EMOD_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EMOD_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2ENUMERAL_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2ENUMERAL_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2ESUBSET_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ESUBSET_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2ESUBSET_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2ESUBSET_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ESUBSET_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2ESUBSET_2E0(A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2ESUBSET_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ESUBSET_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2ESUBSET_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2ESUBSET_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2ESUBSET_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Enum_2ESUC_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enum_2ESUC_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2ESUC_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Eiterate_2ESum_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] : ( c_2Eiterate_2ESum_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Eiterate_2ESum_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Eiterate_2ESum_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)] : ( c_2Eiterate_2ESum_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Eiterate_2ESum_2E0(A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Eiterate_2ESum_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal)] : ( c_2Eiterate_2ESum_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Eiterate_2ESum_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Eiterate_2ESum_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Eiterate_2ESum_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Eiterate_2ESum_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Erealax_2Ereal)] : ( c_2Eiterate_2ESum_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Eiterate_2ESum_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c))] : ( c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,A_27c,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),c_2Epair_2EUNCURRY_2E0(A_27a,A_27b,A_27c),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2Etyop_2Enum_2Enum,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum))] : ( c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Enum_2Enum),c_2Epair_2EUNCURRY_2E0(A_27a,A_27b,tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2Etyop_2Epair_2Eprod_28tyop_2Epair_2Eprod_28A_27a_2CA_27b_29_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))] : ( c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0(A_27a,A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2Etyop_2Erealax_2Ereal,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))] : ( c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E0(A_27a,A_27b,tyop_2Erealax_2Ereal),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27c_20mono_2EA_27d_20mono_2Etyop_2Epair_2Eprod_28A_27b_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))] : ( c_2Epair_2EUNCURRY_2E1(A_27c,A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27d),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0(A_27c,A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27e_20mono_2Etyop_2Epair_2Eprod_28A_27f_2CA_27g_29_20mono_2Etyop_2Epair_2Eprod_28A_27b_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [A_27b: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,X0_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))] : ( c_2Epair_2EUNCURRY_2E1(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g)),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),X0_2E0) ) ).

tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27f_20mono_2EA_27g_20mono_2Etyop_2Epair_2Eprod_28A_27b_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [A_27b: $tType,A_27f: $tType,A_27g: $tType,X0_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))] : ( c_2Epair_2EUNCURRY_2E1(A_27f,A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0(A_27f,A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),X0_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EUNION_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EUNION_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EUNION_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EUNION_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EUNION_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),c_2Epred__set_2EUNION_2E0(A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EUNION_2E2_2Emono_2EA_27e,axiom,
    ! [A_27e: $tType,X0_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EUNION_2E2(A_27e,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool)),c_2Epred__set_2EUNION_2E0(A_27e),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EUNION_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EUNION_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Epred__set_2EUNION_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epred__set_2EUNION_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Epred__set_2EUNION_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(X0_2E0)
        | p(X1_2E0) )
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_5C_2F_2E0,X0_2E0),X1_2E0)) ) ).

tff(arityeq1_2Ec_2Ereal_2Eabs_2E1,axiom,
    ! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Eabs_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Eiterate_2Einf_2E1,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Einf_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,c_2Eiterate_2Einf_2E0,X0_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Eiterate_2E3_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X2_2E0: tyop_2Emin_2Efun(A_27b,A_27a)] : ( c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),A_27a,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),A_27a)),c_2Eiterate_2Eiterate_2E0(A_27b,A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Eiterate_2E3_2Emono_2Etyop_2Enum_2Enum_20mono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] : ( c_2Eiterate_2Eiterate_2E3(tyop_2Enum_2Enum,A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),A_27a,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),A_27a)),c_2Eiterate_2Eiterate_2E0(tyop_2Enum_2Enum,A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Eiterate_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X2_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] : ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),A_27b,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),A_27b),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),A_27b)),c_2Eiterate_2Eiterate_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Eiterate_2E3_2Emono_2EA_27a_20mono_2EA_27c,axiom,
    ! [A_27a: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c)),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X2_2E0: tyop_2Emin_2Efun(A_27a,A_27c)] : ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27c,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,A_27c),A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),A_27c),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),A_27c)),c_2Eiterate_2Eiterate_2E0(A_27a,A_27c),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Eiterate_2E3_2Emono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c)),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X2_2E0: tyop_2Emin_2Efun(A_27b,A_27c)] : ( c_2Eiterate_2Eiterate_2E3(A_27b,A_27c,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,A_27c),A_27c,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27c),A_27c),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27c),A_27c)),c_2Eiterate_2Eiterate_2E0(A_27b,A_27c),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Eiterate_2E3_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2EA_27c,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c)),X1_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),X2_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c)] : ( c_2Eiterate_2Eiterate_2E3(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),A_27c,app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),A_27c),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),A_27c)),c_2Eiterate_2Eiterate_2E0(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2Eiterate_2Eiterate_2E1_2Emono_2EA_27a_20mono_2Etyop_2Enum_2Enum,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))] : ( c_2Eiterate_2Eiterate_2E1(A_27a,tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Enum_2Enum)),c_2Eiterate_2Eiterate_2E0(A_27a,tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eiterate_2Eiterate_2E1_2Emono_2EA_27a_20mono_2Etyop_2Erealax_2Ereal,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] : ( c_2Eiterate_2Eiterate_2E1(A_27a,tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal)),c_2Eiterate_2Eiterate_2E0(A_27a,tyop_2Erealax_2Ereal),X0_2E0) ) ).

tff(arityeq2_2Ec_2Ereal_2Emax_2E2,axiom,
    ! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emax_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ereal_2Emax_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ereal_2Emin_2E2,axiom,
    ! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Emin_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ereal_2Emin_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Eiterate_2Emonoidal_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))] : ( c_2Eiterate_2Emonoidal_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Ebool,c_2Eiterate_2Emonoidal_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eiterate_2Emonoidal_2E1_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] : ( c_2Eiterate_2Emonoidal_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool,c_2Eiterate_2Emonoidal_2E0(A_27b),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eiterate_2Emonoidal_2E1_2Emono_2EA_27c,axiom,
    ! [A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c))] : ( c_2Eiterate_2Emonoidal_2E1(A_27c,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c)),tyop_2Emin_2Ebool,c_2Eiterate_2Emonoidal_2E0(A_27c),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eiterate_2Emonoidal_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))] : ( c_2Eiterate_2Emonoidal_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Ebool,c_2Eiterate_2Emonoidal_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eiterate_2Emonoidal_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] : ( c_2Eiterate_2Emonoidal_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool,c_2Eiterate_2Emonoidal_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eiterate_2Eneutral_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))] : ( c_2Eiterate_2Eneutral_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),A_27a,c_2Eiterate_2Eneutral_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eiterate_2Eneutral_2E1_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] : ( c_2Eiterate_2Eneutral_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),A_27b,c_2Eiterate_2Eneutral_2E0(A_27b),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eiterate_2Eneutral_2E1_2Emono_2EA_27c,axiom,
    ! [A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c))] : ( c_2Eiterate_2Eneutral_2E1(A_27c,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c)),A_27c,c_2Eiterate_2Eneutral_2E0(A_27c),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eiterate_2Eneutral_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))] : ( c_2Eiterate_2Eneutral_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Enum_2Enum,c_2Eiterate_2Eneutral_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).

tff(arityeq1_2Ec_2Eiterate_2Eneutral_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] : ( c_2Eiterate_2Eneutral_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Erealax_2Ereal,c_2Eiterate_2Eneutral_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).

tff(arityeq2_2Ec_2Eiterate_2Ensum_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] : ( c_2Eiterate_2Ensum_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Enum_2Enum),c_2Eiterate_2Ensum_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Eiterate_2Ensum_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)] : ( c_2Eiterate_2Ensum_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),tyop_2Enum_2Enum),c_2Eiterate_2Ensum_2E0(A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Eiterate_2Ensum_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Enum_2Enum)] : ( c_2Eiterate_2Ensum_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Enum_2Enum),tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Enum_2Enum),tyop_2Enum_2Enum),c_2Eiterate_2Ensum_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Eiterate_2Ensum_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] : ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Enum_2Enum),c_2Eiterate_2Ensum_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Eiterate_2Ensum_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Enum_2Enum)] : ( c_2Eiterate_2Ensum_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Enum_2Enum),tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Enum_2Enum),tyop_2Enum_2Enum),c_2Eiterate_2Ensum_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27a)] : ( c_2Ecombin_2Eo_2E2(A_27a,A_27b,A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,A_27b),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,A_27b)),c_2Ecombin_2Eo_2E0(A_27a,A_27b,A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2Etyop_2Enum_2Enum_20mono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27a)] : ( c_2Ecombin_2Eo_2E2(A_27a,tyop_2Enum_2Enum,A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),c_2Ecombin_2Eo_2E0(A_27a,tyop_2Enum_2Enum,A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2Etyop_2Erealax_2Ereal_20mono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27a)] : ( c_2Ecombin_2Eo_2E2(A_27a,tyop_2Erealax_2Ereal,A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),c_2Ecombin_2Eo_2E0(A_27a,tyop_2Erealax_2Ereal,A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2EA_27c_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,A_27c),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] : ( c_2Ecombin_2Eo_2E2(A_27a,A_27c,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),c_2Ecombin_2Eo_2E0(A_27a,A_27c,A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2Etyop_2Enum_2Enum_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] : ( c_2Ecombin_2Eo_2E2(A_27a,tyop_2Enum_2Enum,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),c_2Ecombin_2Eo_2E0(A_27a,tyop_2Enum_2Enum,A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2Etyop_2Erealax_2Ereal_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] : ( c_2Ecombin_2Eo_2E2(A_27a,tyop_2Erealax_2Ereal,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),c_2Ecombin_2Eo_2E0(A_27a,tyop_2Erealax_2Ereal,A_27b),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27h_20mono_2EA_27b_20mono_2EA_27i,axiom,
    ! [A_27b: $tType,A_27h: $tType,A_27i: $tType,X0_2E0: tyop_2Emin_2Efun(A_27i,A_27b),X1_2E0: tyop_2Emin_2Efun(A_27h,A_27i)] : ( c_2Ecombin_2Eo_2E2(A_27h,A_27b,A_27i,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27h,A_27i),tyop_2Emin_2Efun(A_27h,A_27b),app_2E2(tyop_2Emin_2Efun(A_27i,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27h,A_27i),tyop_2Emin_2Efun(A_27h,A_27b)),c_2Ecombin_2Eo_2E0(A_27h,A_27b,A_27i),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ecombin_2Eo_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Eiterate_2Epolynomial__function_2E1,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Eiterate_2Epolynomial__function_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,c_2Eiterate_2Epolynomial__function_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Ereal_2Epow_2E2,axiom,
    ! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Epow_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),c_2Ereal_2Epow_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Erealax_2Ereal__add_2E2,axiom,
    ! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__add_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ereal_2Ereal__gt_2E2,axiom,
    ! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__gt_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal_2Ereal__gt_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Erealax_2Ereal__lt_2E2,axiom,
    ! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Erealax_2Ereal__lt_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ereal_2Ereal__lte_2E2,axiom,
    ! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal_2Ereal__lte_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Erealax_2Ereal__mul_2E2,axiom,
    ! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__mul_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__mul_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Erealax_2Ereal__neg_2E1,axiom,
    ! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__neg_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Ereal_2Ereal__of__num_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__of__num_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Ereal_2Ereal__sub_2E2,axiom,
    ! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ereal_2Ereal__sub_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Ereal_2Esup_2E1,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal_2Esup_2E1(X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,c_2Ereal_2Esup_2E0,X0_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Esupport_2E3_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),X1_2E0: tyop_2Emin_2Efun(A_27b,A_27a),X2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27b,A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),c_2Eiterate_2Esupport_2E0(A_27b,A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Esupport_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),c_2Eiterate_2Esupport_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Esupport_2E3_2Emono_2EA_27c_20mono_2EA_27b,axiom,
    ! [A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),X1_2E0: tyop_2Emin_2Efun(A_27c,A_27b),X2_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27c,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),c_2Eiterate_2Esupport_2E0(A_27c,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Esupport_2E3_2Emono_2EA_27d_20mono_2EA_27b,axiom,
    ! [A_27b: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),X1_2E0: tyop_2Emin_2Efun(A_27d,A_27b),X2_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27d,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27d,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool))),c_2Eiterate_2Esupport_2E0(A_27d,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Esupport_2E3_2Emono_2EA_27e_20mono_2EA_27b,axiom,
    ! [A_27b: $tType,A_27e: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),X1_2E0: tyop_2Emin_2Efun(A_27e,A_27b),X2_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27e,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27e,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),c_2Eiterate_2Esupport_2E0(A_27e,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Esupport_2E3_2Emono_2EA_27f_20mono_2EA_27b,axiom,
    ! [A_27b: $tType,A_27f: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),X1_2E0: tyop_2Emin_2Efun(A_27f,A_27b),X2_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27f,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27f,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool))),c_2Eiterate_2Esupport_2E0(A_27f,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Esupport_2E3_2Emono_2EA_27g_20mono_2EA_27b,axiom,
    ! [A_27b: $tType,A_27g: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),X1_2E0: tyop_2Emin_2Efun(A_27g,A_27b),X2_2E0: tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27g,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27g,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27g,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),c_2Eiterate_2Esupport_2E0(A_27g,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Esupport_2E3_2Emono_2EA_27h_20mono_2EA_27b,axiom,
    ! [A_27b: $tType,A_27h: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),X1_2E0: tyop_2Emin_2Efun(A_27h,A_27b),X2_2E0: tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27h,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27h,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27h,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool))),c_2Eiterate_2Esupport_2E0(A_27h,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Esupport_2E3_2Emono_2EA_27i_20mono_2EA_27b,axiom,
    ! [A_27b: $tType,A_27i: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),X1_2E0: tyop_2Emin_2Efun(A_27i,A_27b),X2_2E0: tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27i,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27i,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27i,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27i,tyop_2Emin_2Ebool))),c_2Eiterate_2Esupport_2E0(A_27i,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Esupport_2E3_2Emono_2EA_27a_20mono_2Etyop_2Enum_2Enum,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),X2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27a,tyop_2Enum_2Enum,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),c_2Eiterate_2Esupport_2E0(A_27a,tyop_2Enum_2Enum),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Eiterate_2Esupport_2E3_2Emono_2EA_27a_20mono_2Etyop_2Erealax_2Ereal,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),X2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27a,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),c_2Eiterate_2Esupport_2E0(A_27a,tyop_2Erealax_2Ereal),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool] :
      ( ~ p(X0_2E0)
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,X0_2E0)) ) ).

tff(thm_2Eiterate_2Esum__def,axiom,
    ! [A_27a: $tType] : ( c_2Eiterate_2ESum_2E0(A_27a) = c_2Eiterate_2Eiterate_2E1(A_27a,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E0) ) ).

tff(thm_2Eiterate_2EITSET__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a),tyop_2Emin_2Ebool)))] :
      ( ! [V2b_2E0: A_27a,V0f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27a)),V3g_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a)] :
          ( p(app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a),tyop_2Emin_2Ebool)),F0_2E0,V2b_2E0),V0f_2E0),V3g_2E0))
        <=> ( ( app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a,V3g_2E0,c_2Epred__set_2EEMPTY_2E0(A_27b)) = V2b_2E0 )
            & ! [V4x_2E0: A_27b,V5s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
                ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V5s_2E0))
               => ( app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a,V3g_2E0,c_2Epred__set_2EINSERT_2E2(A_27b,V4x_2E0,V5s_2E0)) = c_2Ebool_2ECOND_2E3(A_27a,c_2Ebool_2EIN_2E2(A_27b,V4x_2E0,V5s_2E0),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a,V3g_2E0,V5s_2E0),app_2E2(A_27a,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,A_27a),V0f_2E0,V4x_2E0),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a,V3g_2E0,V5s_2E0))) ) ) ) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27a)),V1s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2b_2E0: A_27a] : ( c_2Eiterate_2EITSET_2E3(A_27a,A_27b,V0f_2E0,V1s_2E0,V2b_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a,c_2Emin_2E_40_2E1(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),A_27a),tyop_2Emin_2Ebool)),F0_2E0,V2b_2E0),V0f_2E0)),V1s_2E0) ) ) ).

tff(thm_2Eiterate_2EFINREC__def,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),V1b_2E0: A_27b,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3a_2E0: A_27b] :
          ( p(c_2Eiterate_2EFINREC_2E5(A_27a,A_27b,V0f_2E0,V1b_2E0,V2s_2E0,V3a_2E0,c_2Enum_2E0_2E0))
        <=> ( ( V2s_2E0 = c_2Epred__set_2EEMPTY_2E0(A_27a) )
            & ( V3a_2E0 = V1b_2E0 ) ) )
      & ! [V4f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),V5b_2E0: A_27b,V6s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V7a_2E0: A_27b,V8n_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Eiterate_2EFINREC_2E5(A_27a,A_27b,V4f_2E0,V5b_2E0,V6s_2E0,V7a_2E0,c_2Enum_2ESUC_2E1(V8n_2E0)))
        <=> ? [V9x_2E0: A_27a,V10c_2E0: A_27b] :
              ( p(c_2Ebool_2EIN_2E2(A_27a,V9x_2E0,V6s_2E0))
              & p(c_2Eiterate_2EFINREC_2E5(A_27a,A_27b,V4f_2E0,V5b_2E0,c_2Epred__set_2EDELETE_2E2(A_27a,V6s_2E0,V9x_2E0),V10c_2E0,V8n_2E0))
              & ( V7a_2E0 = app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V4f_2E0,V9x_2E0),V10c_2E0) ) ) ) ) ).

tff(thm_2Eiterate_2Einf,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
      ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
          ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0s_2E0),V1a_2E0))
        <=> ( ! [V2x_2E0: tyop_2Erealax_2Ereal] :
                ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
               => p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2x_2E0)) )
            & ! [V3b_2E0: tyop_2Erealax_2Ereal] :
                ( ! [V4x_2E0: tyop_2Erealax_2Ereal] :
                    ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V0s_2E0))
                   => p(c_2Ereal_2Ereal__lte_2E2(V3b_2E0,V4x_2E0)) )
               => p(c_2Ereal_2Ereal__lte_2E2(V3b_2E0,V1a_2E0)) ) ) )
     => ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Einf_2E1(V0s_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0s_2E0)) ) ) ).

tff(thm_2Eiterate_2Enumseg,axiom,
    ! [F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))))] :
      ( ! [V0m_2E0: tyop_2Enum_2Enum,V2x_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F1_2E0,V0m_2E0),V2x_2E0),V1n_2E0))
        <=> ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V2x_2E0))
            & p(c_2Earithmetic_2E_3C_3D_2E2(V2x_2E0,V1n_2E0)) ) )
     => ( ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2x_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F0_2E0,V0m_2E0),V1n_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V2x_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F1_2E0,V0m_2E0),V2x_2E0),V1n_2E0)) )
       => ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1n_2E0) = c_2Epred__set_2EGSPEC_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F0_2E0,V0m_2E0),V1n_2E0)) ) ) ) ).

tff(thm_2Eiterate_2Eneutral,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
      ( ! [V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),V1x_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F0_2E0,V0op_2E0),V1x_2E0))
        <=> ! [V2y_2E0: A_27a] :
              ( ( app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V1x_2E0),V2y_2E0) = V2y_2E0 )
              & ( app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V2y_2E0),V1x_2E0) = V2y_2E0 ) ) )
     => ! [V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))] : ( c_2Eiterate_2Eneutral_2E1(A_27a,V0op_2E0) = c_2Emin_2E_40_2E1(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F0_2E0,V0op_2E0)) ) ) ).

tff(thm_2Eiterate_2Emonoidal,axiom,
    ! [A_27a: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27a,V0op_2E0))
    <=> ( ! [V1x_2E0: A_27a,V2y_2E0: A_27a] : ( app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V1x_2E0),V2y_2E0) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V2y_2E0),V1x_2E0) )
        & ! [V3x_2E0: A_27a,V4y_2E0: A_27a,V5z_2E0: A_27a] : ( app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V3x_2E0),app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V4y_2E0),V5z_2E0)) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V3x_2E0),V4y_2E0)),V5z_2E0) )
        & ! [V6x_2E0: A_27a] : ( app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,c_2Eiterate_2Eneutral_2E1(A_27a,V0op_2E0)),V6x_2E0) = V6x_2E0 ) ) ) ).

tff(thm_2Eiterate_2Esupport,axiom,
    ! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))))] :
      ( ! [V3x_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
          ( p(app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V2s_2E0),V1f_2E0),V0op_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
            & ( app_2E2(A_27a,A_27b,V1f_2E0,V3x_2E0) != c_2Eiterate_2Eneutral_2E1(A_27b,V0op_2E0) ) ) )
     => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V1f_2E0),V0op_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool))),F1_2E0,V3x_2E0),V2s_2E0),V1f_2E0),V0op_2E0)) )
       => ! [V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27a,A_27b,V0op_2E0,V1f_2E0,V2s_2E0) = c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V2s_2E0),V1f_2E0),V0op_2E0)) ) ) ) ).

tff(thm_2Eiterate_2Eiterate,axiom,
    ! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b))))] :
      ( ! [V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V3x_2E0: A_27a,V4a_2E0: A_27b] : ( app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b))),F0_2E0,V0op_2E0),V2f_2E0),V3x_2E0),V4a_2E0) = app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),V0op_2E0,app_2E2(A_27a,A_27b,V2f_2E0,V3x_2E0)),V4a_2E0) )
     => ! [V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] : ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V1s_2E0,V2f_2E0) = c_2Ebool_2ECOND_2E3(A_27b,c_2Epred__set_2EFINITE_2E1(A_27a,c_2Eiterate_2Esupport_2E3(A_27a,A_27b,V0op_2E0,V2f_2E0,V1s_2E0)),c_2Eiterate_2EITSET_2E3(A_27b,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b))),F0_2E0,V0op_2E0),V2f_2E0),c_2Eiterate_2Esupport_2E3(A_27a,A_27b,V0op_2E0,V2f_2E0,V1s_2E0),c_2Eiterate_2Eneutral_2E1(A_27b,V0op_2E0)),c_2Eiterate_2Eneutral_2E1(A_27b,V0op_2E0)) ) ) ).

tff(thm_2Eiterate_2Ensum__def,axiom,
    ! [A_27a: $tType] : ( c_2Eiterate_2Ensum_2E0(A_27a) = c_2Eiterate_2Eiterate_2E1(A_27a,tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E0) ) ).

tff(thm_2Eiterate_2Epolynomial__function,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
      ( ! [V2c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal,V4i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V3x_2E0),V4i_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2c_2E0,V4i_2E0),c_2Ereal_2Epow_2E2(V3x_2E0,V4i_2E0)) )
     => ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
          ( p(c_2Eiterate_2Epolynomial__function_2E1(V0p_2E0))
        <=> ? [V1m_2E0: tyop_2Enum_2Enum,V2c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
            ! [V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0p_2E0,V3x_2E0) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V1m_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V2c_2E0),V3x_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ESUM__COMBINE__L,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum,V3p_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2n_2E0))
        & p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0))
        & p(c_2Earithmetic_2E_3C_3D_2E2(V2n_2E0,c_2Earithmetic_2E_2B_2E2(V3p_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
     => ( c_2Erealax_2Ereal__add_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,c_2Earithmetic_2E_2D_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0f_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2n_2E0,V3p_2E0),V0f_2E0)) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V3p_2E0),V0f_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__COMBINE__R,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum,V3p_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,c_2Earithmetic_2E_2B_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
        & p(c_2Earithmetic_2E_3C_3D_2E2(V2n_2E0,V3p_2E0)) )
     => ( c_2Erealax_2Ereal__add_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2B_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V3p_2E0),V0f_2E0)) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V3p_2E0),V0f_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__DIFFS__ALT,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3k_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V3k_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V3k_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3k_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0),c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1m_2E0)),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__DIFFS,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3k_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V3k_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3k_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V3k_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0),c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__PARTIAL__PRE,axiom,
    ! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
      ( ! [V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5k_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F1_2E0,V1g_2E0),V0f_2E0),V5k_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V5k_2E0),c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V5k_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V5k_2E0))) )
     => ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4k_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V4k_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4k_2E0),c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4k_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Earithmetic_2E_2D_2E2(V4k_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))) )
       => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V3n_2E0),c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V3n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V3n_2E0)),c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Earithmetic_2E_2D_2E2(V2m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F1_2E0,V1g_2E0),V0f_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ESUM__PARTIAL__SUC,axiom,
    ! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
      ( ! [V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5k_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F1_2E0,V1g_2E0),V0f_2E0),V5k_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Earithmetic_2E_2B_2E2(V5k_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V5k_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V5k_2E0))) )
     => ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4k_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V4k_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4k_2E0),c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Earithmetic_2E_2B_2E2(V4k_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4k_2E0))) )
       => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V3n_2E0),c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V3n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,c_2Earithmetic_2E_2B_2E2(V3n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))),c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V2m_2E0))),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F1_2E0,V1g_2E0),V0f_2E0))),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EREAL__OF__NUM__SUM__NUMSEG,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V3i_2E0) = c_2Ereal_2Ereal__of__num_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V3i_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__of__num_2E1(c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0)) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__PAIR,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V3i_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V3i_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V3i_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V1m_2E0),c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V2n_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0f_2E0) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__CLAUSES__RIGHT,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2n_2E0))
        & p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0)) )
     => ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,c_2Earithmetic_2E_2D_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0f_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V2n_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__CLAUSES__LEFT,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0))
     => ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1m_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2B_2E2(V1m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V2n_2E0),V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__OFFSET__0,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1m_2E0),V3i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V3i_2E0,V1m_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0))
         => ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2E_2D_2E2(V2n_2E0,V1m_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1m_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ESUM__OFFSET,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
      ( ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V0p_2E0: tyop_2Enum_2Enum,V4i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V0p_2E0),V4i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1f_2E0,c_2Earithmetic_2E_2B_2E2(V4i_2E0,V0p_2E0)) )
     => ! [V0p_2E0: tyop_2Enum_2Enum,V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2B_2E2(V2m_2E0,V0p_2E0),c_2Earithmetic_2E_2B_2E2(V3n_2E0,V0p_2E0)),V1f_2E0) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1f_2E0),V0p_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__ADD__SPLIT,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum,V3p_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,c_2Earithmetic_2E_2B_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
     => ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,c_2Earithmetic_2E_2B_2E2(V2n_2E0,V3p_2E0)),V0f_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2B_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2E_2B_2E2(V2n_2E0,V3p_2E0)),V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__SWAP__NUMSEG,axiom,
    ! [F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))))] :
      ( ! [V4f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),V6j_2E0: tyop_2Enum_2Enum,V7i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F2_2E0,V4f_2E0),V6j_2E0),V7i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4f_2E0,V7i_2E0),V6j_2E0) )
     => ( ! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum,V4f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),V6j_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V0a_2E0),V1b_2E0),V4f_2E0),V6j_2E0) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V0a_2E0,V1b_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F2_2E0,V4f_2E0),V6j_2E0)) )
       => ( ! [V2c_2E0: tyop_2Enum_2Enum,V3d_2E0: tyop_2Enum_2Enum,V4f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),V5i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0,V2c_2E0),V3d_2E0),V4f_2E0),V5i_2E0) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2c_2E0,V3d_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4f_2E0,V5i_2E0)) )
         => ! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum,V2c_2E0: tyop_2Enum_2Enum,V3d_2E0: tyop_2Enum_2Enum,V4f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V0a_2E0,V1b_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0,V2c_2E0),V3d_2E0),V4f_2E0)) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2c_2E0,V3d_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0,V0a_2E0),V1b_2E0),V4f_2E0)) ) ) ) ) ).

tff(thm_2Eiterate_2ESUM__CLAUSES__NUMSEG,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( ! [V1m_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V1m_2E0))
        <=> ( V1m_2E0 = c_2Enum_2E0_2E0 ) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
          ( ! [V1m_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,c_2Enum_2E0_2E0),V0f_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V1m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Enum_2E0_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) )
          & ! [V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)),V0f_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)),c_2Erealax_2Ereal__add_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V0f_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,c_2Enum_2ESUC_2E1(V3n_2E0))),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V0f_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ESUM__SING__NUMSEG,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1n_2E0,V1n_2E0),V0f_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V1n_2E0) ) ).

tff(thm_2Eiterate_2ESUM__POS__EQ__0__NUMSEG,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
      ( ( ! [V3p_2E0: tyop_2Enum_2Enum] :
            ( ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V3p_2E0))
              & p(c_2Earithmetic_2E_3C_3D_2E2(V3p_2E0,V2n_2E0)) )
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3p_2E0))) )
        & ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
     => ! [V4p_2E0: tyop_2Enum_2Enum] :
          ( ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V4p_2E0))
            & p(c_2Earithmetic_2E_3C_3D_2E2(V4p_2E0,V2n_2E0)) )
         => ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4p_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).

tff(thm_2Eiterate_2ESUM__POS__LE__NUMSEG,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
      ( ! [V3p_2E0: tyop_2Enum_2Enum] :
          ( ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V3p_2E0))
            & p(c_2Earithmetic_2E_3C_3D_2E2(V3p_2E0,V1n_2E0)) )
         => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V2f_2E0,V3p_2E0))) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1n_2E0),V2f_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__TRIV__NUMSEG,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Eprim__rec_2E_3C_2E2(V2n_2E0,V1m_2E0))
     => ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__EQ__0__NUMSEG,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
      ( ! [V3i_2E0: tyop_2Enum_2Enum] :
          ( ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V3i_2E0))
            & p(c_2Earithmetic_2E_3C_3D_2E2(V3i_2E0,V2n_2E0)) )
         => ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3i_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
     => ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__CONST__NUMSEG,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
      ( ! [V0c_2E0: tyop_2Erealax_2Ereal,V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V3n_2E0) = V0c_2E0 )
     => ! [V0c_2E0: tyop_2Erealax_2Ereal,V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2B_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V1m_2E0)),V0c_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__ABS__NUMSEG,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V3i_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V3i_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0)),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)))) ) ).

tff(thm_2Eiterate_2ESUM__EQ__NUMSEG,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
      ( ! [V4i_2E0: tyop_2Enum_2Enum] :
          ( ( p(c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V4i_2E0))
            & p(c_2Earithmetic_2E_3C_3D_2E2(V4i_2E0,V3n_2E0)) )
         => ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4i_2E0) ) )
     => ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V0f_2E0) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V1g_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__LE__NUMSEG,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
      ( ! [V4i_2E0: tyop_2Enum_2Enum] :
          ( ( p(c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V4i_2E0))
            & p(c_2Earithmetic_2E_3C_3D_2E2(V4i_2E0,V3n_2E0)) )
         => p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4i_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4i_2E0))) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V0f_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V1g_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__SUB__NUMSEG,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V4i_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4i_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4i_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0)) = c_2Ereal_2Ereal__sub_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V0f_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V1g_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__ADD__NUMSEG,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V4i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V4i_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0f_2E0,V4i_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1g_2E0,V4i_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V0f_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V1g_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__CLOSED,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
        & ! [V3x_2E0: tyop_2Erealax_2Ereal,V4y_2E0: tyop_2Erealax_2Ereal] :
            ( ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
              & p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V4y_2E0)) )
           => p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,c_2Erealax_2Ereal__add_2E2(V3x_2E0,V4y_2E0))) )
        & ! [V5a_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V5a_2E0,V2s_2E0))
           => p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V5a_2E0))) ) )
     => p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V1f_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__IMAGE__LE,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(A_27b,tyop_2Erealax_2Ereal,V1g_2E0,app_2E2(A_27a,A_27b,V0f_2E0,V3x_2E0)))) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2ESum_2E2(A_27b,c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V0f_2E0,V2s_2E0),V1g_2E0),c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Erealax_2Ereal,A_27b,V1g_2E0,V0f_2E0)))) ) ).

tff(thm_2Eiterate_2ESUM__LE__INCLUDED,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V4i_2E0: tyop_2Emin_2Efun(A_27b,A_27a)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
        & p(c_2Epred__set_2EFINITE_2E1(A_27b,V3t_2E0))
        & ! [V5y_2E0: A_27b] :
            ( p(c_2Ebool_2EIN_2E2(A_27b,V5y_2E0,V3t_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(A_27b,tyop_2Erealax_2Ereal,V1g_2E0,V5y_2E0))) )
        & ! [V6x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V6x_2E0,V2s_2E0))
           => ? [V7y_2E0: A_27b] :
                ( p(c_2Ebool_2EIN_2E2(A_27b,V7y_2E0,V3t_2E0))
                & ( app_2E2(A_27b,A_27a,V4i_2E0,V7y_2E0) = V6x_2E0 )
                & p(c_2Ereal_2Ereal__lte_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V6x_2E0),app_2E2(A_27b,tyop_2Erealax_2Ereal,V1g_2E0,V7y_2E0))) ) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V0f_2E0),c_2Eiterate_2ESum_2E2(A_27b,V3t_2E0,V1g_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__CASES__1,axiom,
    ! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))))] :
      ( ! [V4x_2E0: A_27a,V3a_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F1_2E0,V4x_2E0),V3a_2E0))
        <=> ( V4x_2E0 = V3a_2E0 ) )
     => ( ! [V3a_2E0: A_27a,V0y_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0,V3a_2E0),V0y_2E0),V1f_2E0),V4x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F1_2E0,V4x_2E0),V3a_2E0),V0y_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0)) )
       => ! [V0y_2E0: tyop_2Erealax_2Ereal,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3a_2E0: A_27a] :
            ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
              & p(c_2Ebool_2EIN_2E2(A_27a,V3a_2E0,V2s_2E0)) )
           => ( c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0,V3a_2E0),V0y_2E0),V1f_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V1f_2E0),c_2Ereal_2Ereal__sub_2E2(V0y_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3a_2E0))) ) ) ) ) ).

tff(thm_2Eiterate_2ESUM__CASES,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))),F4_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F3_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0,V1P_2E0),V2f_2E0),V3g_2E0),V4x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V4x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2f_2E0,V4x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V3g_2E0,V4x_2E0)) )
     => ( ! [V6x_2E0: A_27a,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
            ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F4_2E0,V6x_2E0),V0s_2E0),V1P_2E0))
          <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V6x_2E0,V0s_2E0))
              & ~ p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V6x_2E0)) ) )
       => ( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V6x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F2_2E0,V0s_2E0),V1P_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F4_2E0,V6x_2E0),V0s_2E0),V1P_2E0)) )
         => ( ! [V5x_2E0: A_27a,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
                ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F3_2E0,V5x_2E0),V0s_2E0),V1P_2E0))
              <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V0s_2E0))
                  & p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V5x_2E0)) ) )
           => ( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V0s_2E0),V1P_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F3_2E0,V5x_2E0),V0s_2E0),V1P_2E0)) )
             => ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
                  ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
                 => ( c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0,V1P_2E0),V2f_2E0),V3g_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V0s_2E0),V1P_2E0)),V2f_2E0),c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F2_2E0,V0s_2E0),V1P_2E0)),V3g_2E0)) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2ESUM__BIGUNION__NONZERO,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V6t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V6t_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V6t_2E0,V0f_2E0) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
          ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1s_2E0))
            & ! [V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
                ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0,V1s_2E0))
               => p(c_2Epred__set_2EFINITE_2E1(A_27a,V2t_2E0)) )
            & ! [V3t1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4t2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V5x_2E0: A_27a] :
                ( ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3t1_2E0,V1s_2E0))
                  & p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4t2_2E0,V1s_2E0))
                  & ( V3t1_2E0 != V4t2_2E0 )
                  & p(c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V3t1_2E0))
                  & p(c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V4t2_2E0)) )
               => ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
         => ( c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EBIGUNION_2E1(A_27a,V1s_2E0),V0f_2E0) = c_2Eiterate_2ESum_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ESUM__UNION__NONZERO,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
        & p(c_2Epred__set_2EFINITE_2E1(A_27a,V2t_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,c_2Epred__set_2EINTER_2E2(A_27a,V1s_2E0,V2t_2E0)))
           => ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
     => ( c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EUNION_2E2(A_27a,V1s_2E0,V2t_2E0),V0f_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0),c_2Eiterate_2ESum_2E2(A_27a,V2t_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__INJECTION,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1p_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
           => p(c_2Ebool_2EIN_2E2(A_27a,app_2E2(A_27a,A_27a,V1p_2E0,V3x_2E0),V2s_2E0)) )
        & ! [V4x_2E0: A_27a,V5y_2E0: A_27a] :
            ( ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V2s_2E0))
              & p(c_2Ebool_2EIN_2E2(A_27a,V5y_2E0,V2s_2E0))
              & ( app_2E2(A_27a,A_27a,V1p_2E0,V4x_2E0) = app_2E2(A_27a,A_27a,V1p_2E0,V5y_2E0) ) )
           => ( V4x_2E0 = V5y_2E0 ) ) )
     => ( c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Erealax_2Ereal,A_27a,V0f_2E0,V1p_2E0)) = c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V0f_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__EQ__GENERAL__INVERSES,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V4h_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V5k_2E0: tyop_2Emin_2Efun(A_27b,A_27a)] :
      ( ( ! [V6y_2E0: A_27b] :
            ( p(c_2Ebool_2EIN_2E2(A_27b,V6y_2E0,V1t_2E0))
           => ( p(c_2Ebool_2EIN_2E2(A_27a,app_2E2(A_27b,A_27a,V5k_2E0,V6y_2E0),V0s_2E0))
              & ( app_2E2(A_27a,A_27b,V4h_2E0,app_2E2(A_27b,A_27a,V5k_2E0,V6y_2E0)) = V6y_2E0 ) ) )
        & ! [V7x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V7x_2E0,V0s_2E0))
           => ( p(c_2Ebool_2EIN_2E2(A_27b,app_2E2(A_27a,A_27b,V4h_2E0,V7x_2E0),V1t_2E0))
              & ( app_2E2(A_27b,A_27a,V5k_2E0,app_2E2(A_27a,A_27b,V4h_2E0,V7x_2E0)) = V7x_2E0 )
              & ( app_2E2(A_27b,tyop_2Erealax_2Ereal,V3g_2E0,app_2E2(A_27a,A_27b,V4h_2E0,V7x_2E0)) = app_2E2(A_27a,tyop_2Erealax_2Ereal,V2f_2E0,V7x_2E0) ) ) ) )
     => ( c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,V2f_2E0) = c_2Eiterate_2ESum_2E2(A_27b,V1t_2E0,V3g_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__EQ__GENERAL,axiom,
    ! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4h_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V5y_2E0: A_27b,V6x_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V4h_2E0),V5y_2E0),V6x_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V6x_2E0,V0s_2E0))
            & ( app_2E2(A_27a,A_27b,V4h_2E0,V6x_2E0) = V5y_2E0 ) ) )
     => ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V4h_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
          ( ( ! [V5y_2E0: A_27b] :
                ( p(c_2Ebool_2EIN_2E2(A_27b,V5y_2E0,V1t_2E0))
               => p(c_2Ebool_2E_3F_21_2E1(A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V4h_2E0),V5y_2E0))) )
            & ! [V7x_2E0: A_27a] :
                ( p(c_2Ebool_2EIN_2E2(A_27a,V7x_2E0,V0s_2E0))
               => ( p(c_2Ebool_2EIN_2E2(A_27b,app_2E2(A_27a,A_27b,V4h_2E0,V7x_2E0),V1t_2E0))
                  & ( app_2E2(A_27b,tyop_2Erealax_2Ereal,V3g_2E0,app_2E2(A_27a,A_27b,V4h_2E0,V7x_2E0)) = app_2E2(A_27a,tyop_2Erealax_2Ereal,V2f_2E0,V7x_2E0) ) ) ) )
         => ( c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,V2f_2E0) = c_2Eiterate_2ESum_2E2(A_27b,V1t_2E0,V3g_2E0) ) ) ) ).

tff(thm_2Eiterate_2ESUM__SUM__PRODUCT,axiom,
    ! [A_27a: $tType,A_27b: $tType,F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F3_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))))] :
      ( ! [V2x_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V7i_2E0: A_27a,V8j_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),F2_2E0,V2x_2E0),V7i_2E0),V8j_2E0) = app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V2x_2E0,V7i_2E0),V8j_2E0) )
     => ( ! [V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V2x_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V4i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1t_2E0),V2x_2E0),V4i_2E0) = c_2Eiterate_2ESum_2E2(A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1t_2E0,V4i_2E0),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V2x_2E0,V4i_2E0)) )
       => ( ! [V5i_2E0: A_27a,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V6j_2E0: A_27b,V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
              ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool))),F3_2E0,V5i_2E0),V0s_2E0),V6j_2E0),V1t_2E0))
            <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V5i_2E0,V0s_2E0))
                & p(c_2Ebool_2EIN_2E2(A_27b,V6j_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1t_2E0,V5i_2E0))) ) )
         => ( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V5i_2E0: A_27a,V6j_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))),F1_2E0,V0s_2E0),V1t_2E0),V5i_2E0),V6j_2E0) = c_2Epair_2E_2C_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,c_2Epair_2E_2C_2E2(A_27a,A_27b,V5i_2E0,V6j_2E0),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool))),F3_2E0,V5i_2E0),V0s_2E0),V6j_2E0),V1t_2E0)) )
           => ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V2x_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))] :
                ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
                  & ! [V3i_2E0: A_27a] :
                      ( p(c_2Ebool_2EIN_2E2(A_27a,V3i_2E0,V0s_2E0))
                     => p(c_2Epred__set_2EFINITE_2E1(A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1t_2E0,V3i_2E0))) ) )
               => ( c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1t_2E0),V2x_2E0)) = c_2Eiterate_2ESum_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epred__set_2EGSPEC_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))),F1_2E0,V0s_2E0),V1t_2E0))),c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),F2_2E0,V2x_2E0))) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2ESUM__BIJECTION,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1p_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V4y_2E0: A_27a,V5x_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V2s_2E0),V1p_2E0),V4y_2E0),V5x_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V2s_2E0))
            & ( app_2E2(A_27a,A_27a,V1p_2E0,V5x_2E0) = V4y_2E0 ) ) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1p_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( ( ! [V3x_2E0: A_27a] :
                ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
               => p(c_2Ebool_2EIN_2E2(A_27a,app_2E2(A_27a,A_27a,V1p_2E0,V3x_2E0),V2s_2E0)) )
            & ! [V4y_2E0: A_27a] :
                ( p(c_2Ebool_2EIN_2E2(A_27a,V4y_2E0,V2s_2E0))
               => p(c_2Ebool_2E_3F_21_2E1(A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V2s_2E0),V1p_2E0),V4y_2E0))) ) )
         => ( c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V0f_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Erealax_2Ereal,A_27a,V0f_2E0,V1p_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ESUM__IMAGE__NONZERO,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0d_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V1i_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
        & ! [V3x_2E0: A_27a,V4y_2E0: A_27a] :
            ( ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
              & p(c_2Ebool_2EIN_2E2(A_27a,V4y_2E0,V2s_2E0))
              & ( V3x_2E0 != V4y_2E0 )
              & ( app_2E2(A_27a,A_27b,V1i_2E0,V3x_2E0) = app_2E2(A_27a,A_27b,V1i_2E0,V4y_2E0) ) )
           => ( app_2E2(A_27b,tyop_2Erealax_2Ereal,V0d_2E0,app_2E2(A_27a,A_27b,V1i_2E0,V3x_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
     => ( c_2Eiterate_2ESum_2E2(A_27b,c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V1i_2E0,V2s_2E0),V0d_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Erealax_2Ereal,A_27b,V0d_2E0,V1i_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__SUBSET__SIMPLE,axiom,
    ! [A_27a: $tType,V0u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1v_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1v_2E0))
        & p(c_2Epred__set_2ESUBSET_2E2(A_27a,V0u_2E0,V1v_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,c_2Epred__set_2EDIFF_2E2(A_27a,V1v_2E0,V0u_2E0)))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2f_2E0,V3x_2E0))) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2ESum_2E2(A_27a,V0u_2E0,V2f_2E0),c_2Eiterate_2ESum_2E2(A_27a,V1v_2E0,V2f_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__SUBSET,axiom,
    ! [A_27a: $tType,V0u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1v_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0u_2E0))
        & p(c_2Epred__set_2EFINITE_2E1(A_27a,V1v_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,c_2Epred__set_2EDIFF_2E2(A_27a,V0u_2E0,V1v_2E0)))
           => p(c_2Ereal_2Ereal__lte_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V2f_2E0,V3x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) )
        & ! [V4x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,c_2Epred__set_2EDIFF_2E2(A_27a,V1v_2E0,V0u_2E0)))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2f_2E0,V4x_2E0))) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2ESum_2E2(A_27a,V0u_2E0,V2f_2E0),c_2Eiterate_2ESum_2E2(A_27a,V1v_2E0,V2f_2E0))) ) ).

tff(thm_2Eiterate_2EREAL__OF__NUM__SUM,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2x_2E0) = c_2Ereal_2Ereal__of__num_2E1(app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V2x_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
         => ( c_2Ereal_2Ereal__of__num_2E1(c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,V0f_2E0)) = c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ESUM__GROUP,axiom,
    ! [A_27a: $tType,A_27b: $tType,F2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))))] :
      ( ! [V5x_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V4y_2E0: A_27b] :
          ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F2_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4y_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V2s_2E0))
            & ( app_2E2(A_27a,A_27b,V0f_2E0,V5x_2E0) = V4y_2E0 ) ) )
     => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V4y_2E0: A_27b,V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V4y_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F2_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4y_2E0)) )
       => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))),F0_2E0,V2s_2E0),V0f_2E0),V1g_2E0),V4y_2E0) = c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V4y_2E0)),V1g_2E0) )
         => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
              ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
                & p(c_2Epred__set_2ESUBSET_2E2(A_27b,c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V0f_2E0,V2s_2E0),V3t_2E0)) )
             => ( c_2Eiterate_2ESum_2E2(A_27b,V3t_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))),F0_2E0,V2s_2E0),V0f_2E0),V1g_2E0)) = c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V1g_2E0) ) ) ) ) ) ).

tff(thm_2Eiterate_2ESUM__IMAGE__GEN,axiom,
    ! [A_27a: $tType,A_27b: $tType,F2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))))] :
      ( ! [V4x_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V3y_2E0: A_27b] :
          ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V2s_2E0),V0f_2E0),V3y_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V2s_2E0))
            & ( app_2E2(A_27a,A_27b,V0f_2E0,V4x_2E0) = V3y_2E0 ) ) )
     => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V3y_2E0: A_27b,V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V3y_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V2s_2E0),V0f_2E0),V3y_2E0)) )
       => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))),F0_2E0,V2s_2E0),V0f_2E0),V1g_2E0),V3y_2E0) = c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V3y_2E0)),V1g_2E0) )
         => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
              ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
             => ( c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V1g_2E0) = c_2Eiterate_2ESum_2E2(A_27b,c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V0f_2E0,V2s_2E0),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))),F0_2E0,V2s_2E0),V0f_2E0),V1g_2E0)) ) ) ) ) ) ).

tff(thm_2Eiterate_2ESUM__MULTICOUNT,axiom,
    ! [A_27a: $tType,A_27b: $tType,F4_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))))] :
      ( ! [V7j_2E0: A_27b,V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V6i_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F4_2E0,V7j_2E0),V2t_2E0),V0R_2E0),V6i_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27b,V7j_2E0,V2t_2E0))
            & p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0,V6i_2E0),V7j_2E0)) ) )
     => ( ! [V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V6i_2E0: A_27a,V7j_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F3_2E0,V2t_2E0),V0R_2E0),V6i_2E0),V7j_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,V7j_2E0,app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F4_2E0,V7j_2E0),V2t_2E0),V0R_2E0),V6i_2E0)) )
       => ( ! [V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V6i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V2t_2E0),V0R_2E0),V6i_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Epred__set_2ECARD_2E1(A_27b,c_2Epred__set_2EGSPEC_2E1(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F3_2E0,V2t_2E0),V0R_2E0),V6i_2E0)))) )
         => ( ! [V5i_2E0: A_27a,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V4j_2E0: A_27b] :
                ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F2_2E0,V5i_2E0),V1s_2E0),V0R_2E0),V4j_2E0))
              <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V5i_2E0,V1s_2E0))
                  & p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0,V5i_2E0),V4j_2E0)) ) )
           => ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V4j_2E0: A_27b,V5i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0R_2E0),V4j_2E0),V5i_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V5i_2E0,app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F2_2E0,V5i_2E0),V1s_2E0),V0R_2E0),V4j_2E0)) )
             => ! [V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3k_2E0: tyop_2Enum_2Enum] :
                  ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
                    & p(c_2Epred__set_2EFINITE_2E1(A_27b,V2t_2E0))
                    & ! [V4j_2E0: A_27b] :
                        ( p(c_2Ebool_2EIN_2E2(A_27b,V4j_2E0,V2t_2E0))
                       => ( c_2Epred__set_2ECARD_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0R_2E0),V4j_2E0))) = V3k_2E0 ) ) )
                 => ( c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V2t_2E0),V0R_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2E_2A_2E2(V3k_2E0,c_2Epred__set_2ECARD_2E1(A_27b,V2t_2E0))) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2ESUM__MULTICOUNT__GEN,axiom,
    ! [A_27a: $tType,A_27b: $tType,F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),F5_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F4_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F3_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))))] :
      ( ! [V3k_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),V8i_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),F2_2E0,V3k_2E0),V8i_2E0) = c_2Ereal_2Ereal__of__num_2E1(app_2E2(A_27b,tyop_2Enum_2Enum,V3k_2E0,V8i_2E0)) )
     => ( ! [V7j_2E0: A_27b,V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V6i_2E0: A_27a] :
            ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F5_2E0,V7j_2E0),V2t_2E0),V0R_2E0),V6i_2E0))
          <=> ( p(c_2Ebool_2EIN_2E2(A_27b,V7j_2E0,V2t_2E0))
              & p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0,V6i_2E0),V7j_2E0)) ) )
       => ( ! [V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V6i_2E0: A_27a,V7j_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F4_2E0,V2t_2E0),V0R_2E0),V6i_2E0),V7j_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,V7j_2E0,app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F5_2E0,V7j_2E0),V2t_2E0),V0R_2E0),V6i_2E0)) )
         => ( ! [V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V6i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V2t_2E0),V0R_2E0),V6i_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Epred__set_2ECARD_2E1(A_27b,c_2Epred__set_2EGSPEC_2E1(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F4_2E0,V2t_2E0),V0R_2E0),V6i_2E0)))) )
           => ( ! [V5i_2E0: A_27a,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V4j_2E0: A_27b] :
                  ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F3_2E0,V5i_2E0),V1s_2E0),V0R_2E0),V4j_2E0))
                <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V5i_2E0,V1s_2E0))
                    & p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0,V5i_2E0),V4j_2E0)) ) )
             => ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V4j_2E0: A_27b,V5i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0R_2E0),V4j_2E0),V5i_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V5i_2E0,app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F3_2E0,V5i_2E0),V1s_2E0),V0R_2E0),V4j_2E0)) )
               => ! [V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3k_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)] :
                    ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
                      & p(c_2Epred__set_2EFINITE_2E1(A_27b,V2t_2E0))
                      & ! [V4j_2E0: A_27b] :
                          ( p(c_2Ebool_2EIN_2E2(A_27b,V4j_2E0,V2t_2E0))
                         => ( c_2Epred__set_2ECARD_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0R_2E0),V4j_2E0))) = app_2E2(A_27b,tyop_2Enum_2Enum,V3k_2E0,V4j_2E0) ) ) )
                   => ( c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V2t_2E0),V0R_2E0)) = c_2Eiterate_2ESum_2E2(A_27b,V2t_2E0,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),F2_2E0,V3k_2E0)) ) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2ECARD__EQ__SUM,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
      ( ! [V1x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) )
     => ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
         => ( c_2Ereal_2Ereal__of__num_2E1(c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0)) = c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,F0_2E0) ) ) ) ).

tff(thm_2Eiterate_2ESUM__SUM__RESTRICT,axiom,
    ! [A_27a: $tType,A_27b: $tType,F7_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F5_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F6_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F4_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))))] :
      ( ! [V5y_2E0: A_27b,V3t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V4x_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F7_2E0,V5y_2E0),V3t_2E0),V0R_2E0),V4x_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27b,V5y_2E0,V3t_2E0))
            & p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0,V4x_2E0),V5y_2E0)) ) )
     => ( ! [V3t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V4x_2E0: A_27a,V5y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F5_2E0,V3t_2E0),V0R_2E0),V4x_2E0),V5y_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,V5y_2E0,app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F7_2E0,V5y_2E0),V3t_2E0),V0R_2E0),V4x_2E0)) )
       => ( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V4x_2E0: A_27a,V6y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),F6_2E0,V1f_2E0),V4x_2E0),V6y_2E0) = app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V1f_2E0,V4x_2E0),V6y_2E0) )
         => ( ! [V3t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0,V3t_2E0),V0R_2E0),V1f_2E0),V4x_2E0) = c_2Eiterate_2ESum_2E2(A_27b,c_2Epred__set_2EGSPEC_2E1(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F5_2E0,V3t_2E0),V0R_2E0),V4x_2E0)),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),F6_2E0,V1f_2E0),V4x_2E0)) )
           => ( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V7y_2E0: A_27b,V9x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F3_2E0,V1f_2E0),V7y_2E0),V9x_2E0) = app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V1f_2E0,V9x_2E0),V7y_2E0) )
             => ( ! [V8x_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V7y_2E0: A_27b] :
                    ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F4_2E0,V8x_2E0),V2s_2E0),V0R_2E0),V7y_2E0))
                  <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V8x_2E0,V2s_2E0))
                      & p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0,V8x_2E0),V7y_2E0)) ) )
               => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V7y_2E0: A_27b,V8x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F2_2E0,V2s_2E0),V0R_2E0),V7y_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V8x_2E0,app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F4_2E0,V8x_2E0),V2s_2E0),V0R_2E0),V7y_2E0)) )
                 => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V7y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))),F1_2E0,V2s_2E0),V0R_2E0),V1f_2E0),V7y_2E0) = c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F2_2E0,V2s_2E0),V0R_2E0),V7y_2E0)),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F3_2E0,V1f_2E0),V7y_2E0)) )
                   => ! [V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
                        ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
                          & p(c_2Epred__set_2EFINITE_2E1(A_27b,V3t_2E0)) )
                       => ( c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F0_2E0,V3t_2E0),V0R_2E0),V1f_2E0)) = c_2Eiterate_2ESum_2E2(A_27b,V3t_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal))),F1_2E0,V2s_2E0),V0R_2E0),V1f_2E0)) ) ) ) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2ESUM__RESTRICT__SET,axiom,
    ! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V0P_2E0),V2f_2E0),V4x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V4x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V2f_2E0,V4x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) )
     => ( ! [V3x_2E0: A_27a,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
            ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F2_2E0,V3x_2E0),V1s_2E0),V0P_2E0))
          <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1s_2E0))
              & p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0)) ) )
       => ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V1s_2E0),V0P_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F2_2E0,V3x_2E0),V1s_2E0),V0P_2E0)) )
         => ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] : ( c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V1s_2E0),V0P_2E0)),V2f_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F1_2E0,V0P_2E0),V2f_2E0)) ) ) ) ) ).

tff(thm_2Eiterate_2ESUM__EQ__SUPERSET,axiom,
    ! [A_27a: $tType,V0g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V3t_2E0))
        & p(c_2Epred__set_2ESUBSET_2E2(A_27a,V3t_2E0,V2s_2E0))
        & ! [V4x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V3t_2E0))
           => ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0) = app_2E2(A_27a,tyop_2Erealax_2Ereal,V0g_2E0,V4x_2E0) ) )
        & ! [V5x_2E0: A_27a] :
            ( ( p(c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V2s_2E0))
              & ~ p(c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V3t_2E0)) )
           => ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V5x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
     => ( c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V1f_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V3t_2E0,V0g_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__UNION__EQ,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V3u_2E0))
        & ( c_2Epred__set_2EINTER_2E2(A_27a,V1s_2E0,V2t_2E0) = c_2Epred__set_2EEMPTY_2E0(A_27a) )
        & ( c_2Epred__set_2EUNION_2E2(A_27a,V1s_2E0,V2t_2E0) = V3u_2E0 ) )
     => ( c_2Erealax_2Ereal__add_2E2(c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0),c_2Eiterate_2ESum_2E2(A_27a,V2t_2E0,V0f_2E0)) = c_2Eiterate_2ESum_2E2(A_27a,V3u_2E0,V0f_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__BOUND__LT__GEN,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2b_2E0: tyop_2Erealax_2Ereal] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27a) )
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
           => p(c_2Erealax_2Ereal__lt_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0),c_2Ereal_2E_2F_2E2(V2b_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0))))) ) )
     => p(c_2Erealax_2Ereal__lt_2E2(c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,V1f_2E0),V2b_2E0)) ) ).

tff(thm_2Eiterate_2ESUM__BOUND__LT__ALL,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2b_2E0: tyop_2Erealax_2Ereal] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27a) )
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
           => p(c_2Erealax_2Ereal__lt_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0),V2b_2E0)) ) )
     => p(c_2Erealax_2Ereal__lt_2E2(c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,V1f_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0)),V2b_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__BOUND__LT,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2b_2E0: tyop_2Erealax_2Ereal] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0),V2b_2E0)) )
        & ? [V4x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V0s_2E0))
            & p(c_2Erealax_2Ereal__lt_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V4x_2E0),V2b_2E0)) ) )
     => p(c_2Erealax_2Ereal__lt_2E2(c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,V1f_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0)),V2b_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__ABS__BOUND,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2b_2E0: tyop_2Erealax_2Ereal] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0)),V2b_2E0)) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,V1f_2E0)),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0)),V2b_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__BOUND__GEN,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2b_2E0: tyop_2Erealax_2Ereal] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27a) )
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0),c_2Ereal_2E_2F_2E2(V2b_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0))))) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,V1f_2E0),V2b_2E0)) ) ).

tff(thm_2Eiterate_2ESUM__BOUND,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2b_2E0: tyop_2Erealax_2Ereal] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V3x_2E0),V2b_2E0)) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,V1f_2E0),c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0)),V2b_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__RESTRICT,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
      ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1s_2E0),V0f_2E0),V2x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
         => ( c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1s_2E0),V0f_2E0)) = c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0) ) ) ) ).

tff(thm_2Eiterate_2ESUM__UNION__LZERO,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2v_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2v_2E0))
        & ! [V3x_2E0: A_27a] :
            ( ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1u_2E0))
              & ~ p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2v_2E0)) )
           => ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
     => ( c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EUNION_2E2(A_27a,V1u_2E0,V2v_2E0),V0f_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V2v_2E0,V0f_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__UNION__RZERO,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2v_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1u_2E0))
        & ! [V3x_2E0: A_27a] :
            ( ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2v_2E0))
              & ~ p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1u_2E0)) )
           => ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
     => ( c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EUNION_2E2(A_27a,V1u_2E0,V2v_2E0),V0f_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V1u_2E0,V0f_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__SUPERSET,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2v_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2ESUBSET_2E2(A_27a,V1u_2E0,V2v_2E0))
        & ! [V3x_2E0: A_27a] :
            ( ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2v_2E0))
              & ~ p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1u_2E0)) )
           => ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
     => ( c_2Eiterate_2ESum_2E2(A_27a,V2v_2E0,V0f_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V1u_2E0,V0f_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__IMAGE,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ! [V3x_2E0: A_27a,V4y_2E0: A_27a] :
          ( ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
            & p(c_2Ebool_2EIN_2E2(A_27a,V4y_2E0,V2s_2E0))
            & ( app_2E2(A_27a,A_27b,V0f_2E0,V3x_2E0) = app_2E2(A_27a,A_27b,V0f_2E0,V4y_2E0) ) )
         => ( V3x_2E0 = V4y_2E0 ) )
     => ( c_2Eiterate_2ESum_2E2(A_27b,c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V0f_2E0,V2s_2E0),V1g_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Erealax_2Ereal,A_27b,V1g_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__SWAP,axiom,
    ! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)))] :
      ( ! [V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V3i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2t_2E0),V0f_2E0),V3i_2E0) = c_2Eiterate_2ESum_2E2(A_27b,V2t_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V0f_2E0,V3i_2E0)) )
     => ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V4j_2E0: A_27b,V5i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F2_2E0,V0f_2E0),V4j_2E0),V5i_2E0) = app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V0f_2E0,V5i_2E0),V4j_2E0) )
       => ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V4j_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),F1_2E0,V1s_2E0),V0f_2E0),V4j_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F2_2E0,V0f_2E0),V4j_2E0)) )
         => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
              ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
                & p(c_2Epred__set_2EFINITE_2E1(A_27b,V2t_2E0)) )
             => ( c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2t_2E0),V0f_2E0)) = c_2Eiterate_2ESum_2E2(A_27b,V2t_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),F1_2E0,V1s_2E0),V0f_2E0)) ) ) ) ) ) ).

tff(thm_2Eiterate_2ESUM__DELTA,axiom,
    ! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
      ( ! [V3x_2E0: A_27a,V2a_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F1_2E0,V3x_2E0),V2a_2E0))
        <=> ( V3x_2E0 = V2a_2E0 ) )
     => ( ! [V2a_2E0: A_27a,V0b_2E0: tyop_2Erealax_2Ereal,V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2a_2E0),V0b_2E0),V3x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F1_2E0,V3x_2E0),V2a_2E0),V0b_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) )
       => ! [V0b_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2a_2E0: A_27a] : ( c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V2a_2E0),V0b_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ebool_2EIN_2E2(A_27a,V2a_2E0,V1s_2E0),V0b_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ESUM__SING,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1x_2E0: A_27a] : ( c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EINSERT_2E2(A_27a,V1x_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a)),V0f_2E0) = app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V1x_2E0) ) ).

tff(thm_2Eiterate_2ESUM__DELETE__CASES,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2a_2E0: A_27a] :
      ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
     => ( c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EDELETE_2E2(A_27a,V1s_2E0,V2a_2E0),V0f_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ebool_2EIN_2E2(A_27a,V2a_2E0,V1s_2E0),c_2Ereal_2Ereal__sub_2E2(c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V2a_2E0)),c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__DELETE,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2a_2E0: A_27a] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
        & p(c_2Ebool_2EIN_2E2(A_27a,V2a_2E0,V1s_2E0)) )
     => ( c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EDELETE_2E2(A_27a,V1s_2E0,V2a_2E0),V0f_2E0) = c_2Ereal_2Ereal__sub_2E2(c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V2a_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__ZERO__EXISTS,axiom,
    ! [A_27a: $tType,V0u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
        & ( c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0u_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
     => ( ! [V2i_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V2i_2E0,V1s_2E0))
           => ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V0u_2E0,V2i_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
        | ? [V3j_2E0: A_27a,V4k_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3j_2E0,V1s_2E0))
            & p(c_2Erealax_2Ereal__lt_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0u_2E0,V3j_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0)))
            & p(c_2Ebool_2EIN_2E2(A_27a,V4k_2E0,V1s_2E0))
            & p(c_2Ereal_2Ereal__gt_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0u_2E0,V4k_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0))) ) ) ) ).

tff(thm_2Eiterate_2ESUM__POS__EQ__0,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
        & ! [V2x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0))) )
        & ( c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
     => ! [V3x_2E0: A_27a] :
          ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1s_2E0))
         => ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ).

tff(thm_2Eiterate_2ESUM__POS__BOUND,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1b_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0))) )
        & p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V0f_2E0),V1b_2E0)) )
     => ! [V4x_2E0: A_27a] :
          ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V2s_2E0))
         => p(c_2Ereal_2Ereal__lte_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),V1b_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__POS__LE,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ! [V2x_2E0: A_27a] :
          ( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
         => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0))) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__CONST,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
      ( ! [V0c_2E0: tyop_2Erealax_2Ereal,V2n_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V2n_2E0) = V0c_2E0 )
     => ! [V0c_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
         => ( c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Epred__set_2ECARD_2E1(A_27a,V1s_2E0)),V0c_2E0) ) ) ) ).

tff(thm_2Eiterate_2ESUM__ABS__LE,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0)),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0))) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V0f_2E0)),c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V1g_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__ABS,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2x_2E0) = c_2Ereal_2Eabs_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
         => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0)),c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)))) ) ) ).

tff(thm_2Eiterate_2ESUM__EQ,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ! [V3x_2E0: A_27a] :
          ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
         => ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0) ) )
     => ( c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V0f_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V1g_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__POS__LT__ALL,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27a) )
        & ! [V2i_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V2i_2E0,V0s_2E0))
           => p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1f_2E0,V2i_2E0))) ) )
     => p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,V1f_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__POS__LT,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
        & ! [V2x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0))) )
        & ? [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1s_2E0))
            & p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0))) ) )
     => p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__LT__ALL,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
        & ( V2s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27a) )
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
           => p(c_2Erealax_2Ereal__lt_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0))) ) )
     => p(c_2Erealax_2Ereal__lt_2E2(c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V0f_2E0),c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V1g_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__LT,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0))) )
        & ? [V4x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V2s_2E0))
            & p(c_2Erealax_2Ereal__lt_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V4x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V4x_2E0))) ) )
     => p(c_2Erealax_2Ereal__lt_2E2(c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V0f_2E0),c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V1g_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__LE,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0))) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V0f_2E0),c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V1g_2E0))) ) ).

tff(thm_2Eiterate_2ESUM__SUB,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V3x_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
         => ( c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0)) = c_2Ereal_2Ereal__sub_2E2(c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V0f_2E0),c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V1g_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ESUM__NEG,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V2x_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)) = c_2Erealax_2Ereal__neg_2E1(c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__RMUL,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1c_2E0: tyop_2Erealax_2Ereal,V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1c_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V1c_2E0) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1c_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1c_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V0f_2E0),V1c_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__LMUL,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
      ( ! [V1c_2E0: tyop_2Erealax_2Ereal,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V0f_2E0),V3x_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1c_2E0: tyop_2Erealax_2Ereal,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V0f_2E0)) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__0,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
      ( ! [V1n_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,F0_2E0,V1n_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
     => ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,F0_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__EQ__0,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ! [V2x_2E0: A_27a] :
          ( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
         => ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
     => ( c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ).

tff(thm_2Eiterate_2ESUM__ADD__GEN,axiom,
    ! [A_27a: $tType,F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))),F4_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F3_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F2_2E0,V0f_2E0),V1g_2E0),V5x_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V5x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V5x_2E0)) )
     => ( ! [V4x_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
            ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)),F4_2E0,V4x_2E0),V2s_2E0),V1g_2E0))
          <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V2s_2E0))
              & ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V4x_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
       => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V2s_2E0),V1g_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)),F4_2E0,V4x_2E0),V2s_2E0),V1g_2E0)) )
         => ( ! [V3x_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
                ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)),F3_2E0,V3x_2E0),V2s_2E0),V0f_2E0))
              <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
                  & ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
           => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V2s_2E0),V0f_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)),F3_2E0,V3x_2E0),V2s_2E0),V0f_2E0)) )
             => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
                  ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V2s_2E0),V0f_2E0))))
                    & p(c_2Epred__set_2EFINITE_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V2s_2E0),V1g_2E0)))) )
                 => ( c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F2_2E0,V0f_2E0),V1g_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V0f_2E0),c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V1g_2E0)) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2ESUM__ADD,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0),V3x_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
         => ( c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),F0_2E0,V0f_2E0),V1g_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V0f_2E0),c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V1g_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ESUM__SUPPORT,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2ESum_2E2(A_27a,c_2Eiterate_2Esupport_2E3(A_27a,tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E0,V0f_2E0,V1s_2E0),V0f_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0) ) ).

tff(thm_2Eiterate_2ESUM__INCL__EXCL,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
        & p(c_2Epred__set_2EFINITE_2E1(A_27a,V1t_2E0)) )
     => ( c_2Erealax_2Ereal__add_2E2(c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,V2f_2E0),c_2Eiterate_2ESum_2E2(A_27a,V1t_2E0,V2f_2E0)) = c_2Erealax_2Ereal__add_2E2(c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EUNION_2E2(A_27a,V0s_2E0,V1t_2E0),V2f_2E0),c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EINTER_2E2(A_27a,V0s_2E0,V1t_2E0),V2f_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__DIFF,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
        & p(c_2Epred__set_2ESUBSET_2E2(A_27a,V2t_2E0,V1s_2E0)) )
     => ( c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EDIFF_2E2(A_27a,V1s_2E0,V2t_2E0),V0f_2E0) = c_2Ereal_2Ereal__sub_2E2(c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0),c_2Eiterate_2ESum_2E2(A_27a,V2t_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__UNION,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
        & p(c_2Epred__set_2EFINITE_2E1(A_27a,V2t_2E0))
        & p(c_2Epred__set_2EDISJOINT_2E2(A_27a,V1s_2E0,V2t_2E0)) )
     => ( c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EUNION_2E2(A_27a,V1s_2E0,V2t_2E0),V0f_2E0) = c_2Erealax_2Ereal__add_2E2(c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0),c_2Eiterate_2ESum_2E2(A_27a,V2t_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUM__CLAUSES,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] : ( c_2Eiterate_2ESum_2E2(A_27a,c_2Epred__set_2EEMPTY_2E0(A_27a),V0f_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
      & ! [V1x_2E0: A_27b,V2f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V3s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V3s_2E0))
         => ( c_2Eiterate_2ESum_2E2(A_27b,c_2Epred__set_2EINSERT_2E2(A_27b,V1x_2E0,V3s_2E0),V2f_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,c_2Ebool_2EIN_2E2(A_27b,V1x_2E0,V3s_2E0),c_2Eiterate_2ESum_2E2(A_27b,V3s_2E0,V2f_2E0),c_2Erealax_2Ereal__add_2E2(app_2E2(A_27b,tyop_2Erealax_2Ereal,V2f_2E0,V1x_2E0),c_2Eiterate_2ESum_2E2(A_27b,V3s_2E0,V2f_2E0))) ) ) ) ).

tff(thm_2Eiterate_2ESUM__DEGENERATE,axiom,
    ! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V2x_2E0: A_27a,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)] :
          ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)),F1_2E0,V2x_2E0),V1s_2E0),V0f_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
            & ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V2x_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) )
     => ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V1s_2E0),V0f_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V2x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)),F1_2E0,V2x_2E0),V1s_2E0),V0f_2E0)) )
       => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
            ( ~ p(c_2Epred__set_2EFINITE_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V1s_2E0),V0f_2E0))))
           => ( c_2Eiterate_2ESum_2E2(A_27a,V1s_2E0,V0f_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ) ).

tff(thm_2Eiterate_2EMONOIDAL__REAL__MUL,axiom,
    p(c_2Eiterate_2Emonoidal_2E1(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E0)) ).

tff(thm_2Eiterate_2EMONOIDAL__REAL__ADD,axiom,
    p(c_2Eiterate_2Emonoidal_2E1(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E0)) ).

tff(thm_2Eiterate_2ENEUTRAL__REAL__MUL,axiom,
    c_2Eiterate_2Eneutral_2E1(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ).

tff(thm_2Eiterate_2ENEUTRAL__REAL__ADD,axiom,
    c_2Eiterate_2Eneutral_2E1(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ).

tff(thm_2Eiterate_2ECARD__BIGUNION,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0s_2E0))
        & ! [V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0,V0s_2E0))
           => p(c_2Epred__set_2EFINITE_2E1(A_27a,V1t_2E0)) )
        & ! [V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
            ( ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0,V0s_2E0))
              & p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3u_2E0,V0s_2E0))
              & ( V2t_2E0 != V3u_2E0 ) )
           => ( c_2Epred__set_2EINTER_2E2(A_27a,V2t_2E0,V3u_2E0) = c_2Epred__set_2EEMPTY_2E0(A_27a) ) ) )
     => ( c_2Epred__set_2ECARD_2E1(A_27a,c_2Epred__set_2EBIGUNION_2E1(A_27a,V0s_2E0)) = c_2Eiterate_2Ensum_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0s_2E0,c_2Epred__set_2ECARD_2E0(A_27a)) ) ) ).

tff(thm_2Eiterate_2EINF__UNIQUE__FINITE,axiom,
    ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1s_2E0))
        & ( V1s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ( ( c_2Eiterate_2Einf_2E1(V1s_2E0) = V0a_2E0 )
      <=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0a_2E0,V1s_2E0))
          & ! [V2y_2E0: tyop_2Erealax_2Ereal] :
              ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V1s_2E0))
             => p(c_2Ereal_2Ereal__lte_2E2(V0a_2E0,V2y_2E0)) ) ) ) ) ).

tff(thm_2Eiterate_2ESUP__UNIQUE__FINITE,axiom,
    ! [V0a_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1s_2E0))
        & ( V1s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ( ( c_2Ereal_2Esup_2E1(V1s_2E0) = V0a_2E0 )
      <=> ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V0a_2E0,V1s_2E0))
          & ! [V2y_2E0: tyop_2Erealax_2Ereal] :
              ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2y_2E0,V1s_2E0))
             => p(c_2Ereal_2Ereal__lte_2E2(V2y_2E0,V0a_2E0)) ) ) ) ) ).

tff(thm_2Eiterate_2EREAL__SUP__LT__FINITE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Esup_2E1(V0s_2E0),V1a_2E0))
      <=> ! [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
           => p(c_2Erealax_2Ereal__lt_2E2(V2x_2E0,V1a_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EREAL__LT__SUP__FINITE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ( p(c_2Erealax_2Ereal__lt_2E2(V1a_2E0,c_2Ereal_2Esup_2E1(V0s_2E0)))
      <=> ? [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
            & p(c_2Erealax_2Ereal__lt_2E2(V1a_2E0,V2x_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EREAL__SUP__LE__FINITE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Esup_2E1(V0s_2E0),V1a_2E0))
      <=> ! [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V2x_2E0,V1a_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EREAL__LE__SUP__FINITE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,c_2Ereal_2Esup_2E1(V0s_2E0)))
      <=> ? [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
            & p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2x_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ESUP__FINITE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Ereal_2Esup_2E1(V0s_2E0),V0s_2E0))
        & ! [V1x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V1x_2E0,c_2Ereal_2Esup_2E1(V0s_2E0))) ) ) ) ).

tff(thm_2Eiterate_2ESUP__FINITE__LEMMA,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ? [V1b_2E0: tyop_2Erealax_2Ereal] :
          ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1b_2E0,V0s_2E0))
          & ! [V2x_2E0: tyop_2Erealax_2Ereal] :
              ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
             => p(c_2Ereal_2Ereal__lte_2E2(V2x_2E0,V1b_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ESUP,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
      ( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & ? [V1b_2E0: tyop_2Erealax_2Ereal] :
          ! [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V2x_2E0,V1b_2E0)) ) )
     => ( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,c_2Ereal_2Esup_2E1(V0s_2E0))) )
        & ! [V4b_2E0: tyop_2Erealax_2Ereal] :
            ( ! [V5x_2E0: tyop_2Erealax_2Ereal] :
                ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V0s_2E0))
               => p(c_2Ereal_2Ereal__lte_2E2(V5x_2E0,V4b_2E0)) )
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Esup_2E1(V0s_2E0),V4b_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ESUP__EQ,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
      ( ! [V2b_2E0: tyop_2Erealax_2Ereal] :
          ( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
              ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
             => p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2b_2E0)) )
        <=> ! [V4x_2E0: tyop_2Erealax_2Ereal] :
              ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1t_2E0))
             => p(c_2Ereal_2Ereal__lte_2E2(V4x_2E0,V2b_2E0)) ) )
     => ( c_2Ereal_2Esup_2E1(V0s_2E0) = c_2Ereal_2Esup_2E1(V1t_2E0) ) ) ).

tff(thm_2Eiterate_2Esup__alt,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
      ( ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
          ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0s_2E0),V1a_2E0))
        <=> ( ! [V2x_2E0: tyop_2Erealax_2Ereal] :
                ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
               => p(c_2Ereal_2Ereal__lte_2E2(V2x_2E0,V1a_2E0)) )
            & ! [V3b_2E0: tyop_2Erealax_2Ereal] :
                ( ! [V4x_2E0: tyop_2Erealax_2Ereal] :
                    ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V0s_2E0))
                   => p(c_2Ereal_2Ereal__lte_2E2(V4x_2E0,V3b_2E0)) )
               => p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V3b_2E0)) ) ) )
     => ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Ereal_2Esup_2E1(V0s_2E0) = c_2Emin_2E_40_2E1(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),F0_2E0,V0s_2E0)) ) ) ).

tff(thm_2Eiterate_2EREAL__COMPLETE,axiom,
    ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
      ( ( ? [V1x_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V1x_2E0))
        & ? [V2M_2E0: tyop_2Erealax_2Ereal] :
          ! [V3x_2E0: tyop_2Erealax_2Ereal] :
            ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2M_2E0)) ) )
     => ? [V4M_2E0: tyop_2Erealax_2Ereal] :
          ( ! [V5x_2E0: tyop_2Erealax_2Ereal] :
              ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V5x_2E0))
             => p(c_2Ereal_2Ereal__lte_2E2(V5x_2E0,V4M_2E0)) )
          & ! [V6M_27_2E0: tyop_2Erealax_2Ereal] :
              ( ! [V7x_2E0: tyop_2Erealax_2Ereal] :
                  ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V0P_2E0,V7x_2E0))
                 => p(c_2Ereal_2Ereal__lte_2E2(V7x_2E0,V6M_27_2E0)) )
             => p(c_2Ereal_2Ereal__lte_2E2(V4M_2E0,V6M_27_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ECHOOSE__SUBSET,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
     => ! [V1n_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0)))
         => ? [V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
              ( p(c_2Epred__set_2ESUBSET_2E2(A_27a,V2t_2E0,V0s_2E0))
              & p(c_2Ecardinal_2EHAS__SIZE_2E2(A_27a,V2t_2E0,V1n_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ECHOOSE__SUBSET__STRONG,axiom,
    ! [A_27a: $tType,V0n_2E0: tyop_2Enum_2Enum,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
       => p(c_2Earithmetic_2E_3C_3D_2E2(V0n_2E0,c_2Epred__set_2ECARD_2E1(A_27a,V1s_2E0))) )
     => ? [V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2ESUBSET_2E2(A_27a,V2t_2E0,V1s_2E0))
          & p(c_2Ecardinal_2EHAS__SIZE_2E2(A_27a,V2t_2E0,V0n_2E0)) ) ) ).

tff(thm_2Eiterate_2ESET__PROVE__CASES,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
      ( ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a)))
        & ! [V1a_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
            ( ~ p(c_2Ebool_2EIN_2E2(A_27a,V1a_2E0,V2s_2E0))
           => p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,c_2Epred__set_2EINSERT_2E2(A_27a,V1a_2E0,V2s_2E0))) ) )
     => ! [V3s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,V0P_2E0,V3s_2E0)) ) ).

tff(thm_2Eiterate_2Ereal__INFINITE,axiom,
    ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).

tff(thm_2Eiterate_2EFINITE__REAL__INTERVAL,axiom,
    ! [F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F3_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F11_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F7_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F10_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F5_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F9_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F6_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F8_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F4_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
      ( ! [V2a_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V2a_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V3x_2E0)) )
     => ( ! [V6b_2E0: tyop_2Erealax_2Ereal,V7x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F3_2E0,V6b_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V7x_2E0,c_2Ereal_2Ereal__lte_2E2(V7x_2E0,V6b_2E0)) )
       => ( ! [V0a_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V1x_2E0,c_2Erealax_2Ereal__lt_2E2(V0a_2E0,V1x_2E0)) )
         => ( ! [V4b_2E0: tyop_2Erealax_2Ereal,V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V4b_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5x_2E0,c_2Erealax_2Ereal__lt_2E2(V5x_2E0,V4b_2E0)) )
           => ( ! [V17a_2E0: tyop_2Erealax_2Ereal,V19x_2E0: tyop_2Erealax_2Ereal,V18b_2E0: tyop_2Erealax_2Ereal] :
                  ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F11_2E0,V17a_2E0),V19x_2E0),V18b_2E0))
                <=> ( p(c_2Ereal_2Ereal__lte_2E2(V17a_2E0,V19x_2E0))
                    & p(c_2Ereal_2Ereal__lte_2E2(V19x_2E0,V18b_2E0)) ) )
             => ( ! [V17a_2E0: tyop_2Erealax_2Ereal,V18b_2E0: tyop_2Erealax_2Ereal,V19x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F7_2E0,V17a_2E0),V18b_2E0),V19x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V19x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F11_2E0,V17a_2E0),V19x_2E0),V18b_2E0)) )
               => ( ! [V11a_2E0: tyop_2Erealax_2Ereal,V13x_2E0: tyop_2Erealax_2Ereal,V12b_2E0: tyop_2Erealax_2Ereal] :
                      ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F10_2E0,V11a_2E0),V13x_2E0),V12b_2E0))
                    <=> ( p(c_2Ereal_2Ereal__lte_2E2(V11a_2E0,V13x_2E0))
                        & p(c_2Erealax_2Ereal__lt_2E2(V13x_2E0,V12b_2E0)) ) )
                 => ( ! [V11a_2E0: tyop_2Erealax_2Ereal,V12b_2E0: tyop_2Erealax_2Ereal,V13x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F5_2E0,V11a_2E0),V12b_2E0),V13x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V13x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F10_2E0,V11a_2E0),V13x_2E0),V12b_2E0)) )
                   => ( ! [V14a_2E0: tyop_2Erealax_2Ereal,V16x_2E0: tyop_2Erealax_2Ereal,V15b_2E0: tyop_2Erealax_2Ereal] :
                          ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F9_2E0,V14a_2E0),V16x_2E0),V15b_2E0))
                        <=> ( p(c_2Erealax_2Ereal__lt_2E2(V14a_2E0,V16x_2E0))
                            & p(c_2Ereal_2Ereal__lte_2E2(V16x_2E0,V15b_2E0)) ) )
                     => ( ! [V14a_2E0: tyop_2Erealax_2Ereal,V15b_2E0: tyop_2Erealax_2Ereal,V16x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F6_2E0,V14a_2E0),V15b_2E0),V16x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V16x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F9_2E0,V14a_2E0),V16x_2E0),V15b_2E0)) )
                       => ( ! [V8a_2E0: tyop_2Erealax_2Ereal,V10x_2E0: tyop_2Erealax_2Ereal,V9b_2E0: tyop_2Erealax_2Ereal] :
                              ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F8_2E0,V8a_2E0),V10x_2E0),V9b_2E0))
                            <=> ( p(c_2Erealax_2Ereal__lt_2E2(V8a_2E0,V10x_2E0))
                                & p(c_2Erealax_2Ereal__lt_2E2(V10x_2E0,V9b_2E0)) ) )
                         => ( ! [V8a_2E0: tyop_2Erealax_2Ereal,V9b_2E0: tyop_2Erealax_2Ereal,V10x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F4_2E0,V8a_2E0),V9b_2E0),V10x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V10x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F8_2E0,V8a_2E0),V10x_2E0),V9b_2E0)) )
                           => ( ! [V0a_2E0: tyop_2Erealax_2Ereal] : ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F0_2E0,V0a_2E0))))
                              & ! [V2a_2E0: tyop_2Erealax_2Ereal] : ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V2a_2E0))))
                              & ! [V4b_2E0: tyop_2Erealax_2Ereal] : ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V4b_2E0))))
                              & ! [V6b_2E0: tyop_2Erealax_2Ereal] : ~ p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F3_2E0,V6b_2E0))))
                              & ! [V8a_2E0: tyop_2Erealax_2Ereal,V9b_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F4_2E0,V8a_2E0),V9b_2E0))) = c_2Ereal_2Ereal__lte_2E2(V9b_2E0,V8a_2E0) )
                              & ! [V11a_2E0: tyop_2Erealax_2Ereal,V12b_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F5_2E0,V11a_2E0),V12b_2E0))) = c_2Ereal_2Ereal__lte_2E2(V12b_2E0,V11a_2E0) )
                              & ! [V14a_2E0: tyop_2Erealax_2Ereal,V15b_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F6_2E0,V14a_2E0),V15b_2E0))) = c_2Ereal_2Ereal__lte_2E2(V15b_2E0,V14a_2E0) )
                              & ! [V17a_2E0: tyop_2Erealax_2Ereal,V18b_2E0: tyop_2Erealax_2Ereal] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F7_2E0,V17a_2E0),V18b_2E0))) = c_2Ereal_2Ereal__lte_2E2(V18b_2E0,V17a_2E0) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2EFINITE__RESTRICT,axiom,
    ! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V2x_2E0: A_27a,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F1_2E0,V2x_2E0),V0s_2E0),V1P_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V0s_2E0))
            & p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V2x_2E0)) ) )
     => ( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V1P_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V2x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F1_2E0,V2x_2E0),V0s_2E0),V1P_2E0)) )
       => ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
            ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
           => p(c_2Epred__set_2EFINITE_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V1P_2E0)))) ) ) ) ).

tff(thm_2Eiterate_2ESUBSET__RESTRICT,axiom,
    ! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V2x_2E0: A_27a,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F1_2E0,V2x_2E0),V0s_2E0),V1P_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V0s_2E0))
            & p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V2x_2E0)) ) )
     => ( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V1P_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V2x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F1_2E0,V2x_2E0),V0s_2E0),V1P_2E0)) )
       => ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V1P_2E0)),V0s_2E0)) ) ) ).

tff(thm_2Eiterate_2ECARD__UNION__EQ,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2u_2E0))
        & ( c_2Epred__set_2EINTER_2E2(A_27a,V0s_2E0,V1t_2E0) = c_2Epred__set_2EEMPTY_2E0(A_27a) )
        & ( c_2Epred__set_2EUNION_2E2(A_27a,V0s_2E0,V1t_2E0) = V2u_2E0 ) )
     => ( c_2Earithmetic_2E_2B_2E2(c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0),c_2Epred__set_2ECARD_2E1(A_27a,V1t_2E0)) = c_2Epred__set_2ECARD_2E1(A_27a,V2u_2E0) ) ) ).

tff(thm_2Eiterate_2EFINITE__RECURSION,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),V1b_2E0: A_27b] :
      ( ! [V2x_2E0: A_27a,V3y_2E0: A_27a,V4s_2E0: A_27b] :
          ( ( V2x_2E0 != V3y_2E0 )
         => ( app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V2x_2E0),app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V3y_2E0),V4s_2E0)) = app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V3y_2E0),app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V2x_2E0),V4s_2E0)) ) )
     => ( ( c_2Eiterate_2EITSET_2E3(A_27b,A_27a,V0f_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a),V1b_2E0) = V1b_2E0 )
        & ! [V5x_2E0: A_27a,V6s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
            ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V6s_2E0))
           => ( c_2Eiterate_2EITSET_2E3(A_27b,A_27a,V0f_2E0,c_2Epred__set_2EINSERT_2E2(A_27a,V5x_2E0,V6s_2E0),V1b_2E0) = c_2Ebool_2ECOND_2E3(A_27b,c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V6s_2E0),c_2Eiterate_2EITSET_2E3(A_27b,A_27a,V0f_2E0,V6s_2E0,V1b_2E0),app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V5x_2E0),c_2Eiterate_2EITSET_2E3(A_27b,A_27a,V0f_2E0,V6s_2E0,V1b_2E0))) ) ) ) ) ).

tff(thm_2Eiterate_2ESET__RECURSION__LEMMA,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),V1b_2E0: A_27b] :
      ( ! [V2x_2E0: A_27a,V3y_2E0: A_27a,V4s_2E0: A_27b] :
          ( ( V2x_2E0 != V3y_2E0 )
         => ( app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V2x_2E0),app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V3y_2E0),V4s_2E0)) = app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V3y_2E0),app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V2x_2E0),V4s_2E0)) ) )
     => ? [V5g_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27b)] :
          ( ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27b,V5g_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a)) = V1b_2E0 )
          & ! [V6x_2E0: A_27a,V7s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
              ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V7s_2E0))
             => ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27b,V5g_2E0,c_2Epred__set_2EINSERT_2E2(A_27a,V6x_2E0,V7s_2E0)) = c_2Ebool_2ECOND_2E3(A_27b,c_2Ebool_2EIN_2E2(A_27a,V6x_2E0,V7s_2E0),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27b,V5g_2E0,V7s_2E0),app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V6x_2E0),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27b,V5g_2E0,V7s_2E0))) ) ) ) ) ).

tff(thm_2Eiterate_2EFINREC__FUN,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),V1b_2E0: A_27b] :
      ( ! [V2x_2E0: A_27a,V3y_2E0: A_27a,V4s_2E0: A_27b] :
          ( ( V2x_2E0 != V3y_2E0 )
         => ( app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V2x_2E0),app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V3y_2E0),V4s_2E0)) = app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V3y_2E0),app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V2x_2E0),V4s_2E0)) ) )
     => ? [V5g_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27b)] :
          ( ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27b,V5g_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a)) = V1b_2E0 )
          & ! [V6s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V7x_2E0: A_27a] :
              ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V6s_2E0))
                & p(c_2Ebool_2EIN_2E2(A_27a,V7x_2E0,V6s_2E0)) )
             => ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27b,V5g_2E0,V6s_2E0) = app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V7x_2E0),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27b,V5g_2E0,c_2Epred__set_2EDELETE_2E2(A_27a,V6s_2E0,V7x_2E0))) ) ) ) ) ).

tff(thm_2Eiterate_2EFINREC__FUN__LEMMA,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)))] :
      ( ( ! [V2s_2E0: A_27a] :
            ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V2s_2E0))
           => ? [V3a_2E0: A_27b,V4n_2E0: A_27c] : p(app_2E2(A_27c,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),V1R_2E0,V2s_2E0),V3a_2E0),V4n_2E0)) )
        & ! [V5n1_2E0: A_27c,V6n2_2E0: A_27c,V7s_2E0: A_27a,V8a1_2E0: A_27b,V9a2_2E0: A_27b] :
            ( ( p(app_2E2(A_27c,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),V1R_2E0,V7s_2E0),V8a1_2E0),V5n1_2E0))
              & p(app_2E2(A_27c,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),V1R_2E0,V7s_2E0),V9a2_2E0),V6n2_2E0)) )
           => ( ( V8a1_2E0 = V9a2_2E0 )
              & ( V5n1_2E0 = V6n2_2E0 ) ) ) )
     => ? [V10f_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
        ! [V11s_2E0: A_27a,V12a_2E0: A_27b] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V11s_2E0))
         => ( ? [V13n_2E0: A_27c] : p(app_2E2(A_27c,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),V1R_2E0,V11s_2E0),V12a_2E0),V13n_2E0))
          <=> ( app_2E2(A_27a,A_27b,V10f_2E0,V11s_2E0) = V12a_2E0 ) ) ) ) ).

tff(thm_2Eiterate_2EFINREC__EXISTS__LEMMA,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),V1b_2E0: A_27b,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
     => ? [V3a_2E0: A_27b,V4n_2E0: tyop_2Enum_2Enum] : p(c_2Eiterate_2EFINREC_2E5(A_27a,A_27b,V0f_2E0,V1b_2E0,V2s_2E0,V3a_2E0,V4n_2E0)) ) ).

tff(thm_2Eiterate_2EFINREC__UNIQUE__LEMMA,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),V1b_2E0: A_27b] :
      ( ! [V2x_2E0: A_27a,V3y_2E0: A_27a,V4s_2E0: A_27b] :
          ( ( V2x_2E0 != V3y_2E0 )
         => ( app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V2x_2E0),app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V3y_2E0),V4s_2E0)) = app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V3y_2E0),app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V2x_2E0),V4s_2E0)) ) )
     => ! [V5n1_2E0: tyop_2Enum_2Enum,V6n2_2E0: tyop_2Enum_2Enum,V7s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V8a1_2E0: A_27b,V9a2_2E0: A_27b] :
          ( ( p(c_2Eiterate_2EFINREC_2E5(A_27a,A_27b,V0f_2E0,V1b_2E0,V7s_2E0,V8a1_2E0,V5n1_2E0))
            & p(c_2Eiterate_2EFINREC_2E5(A_27a,A_27b,V0f_2E0,V1b_2E0,V7s_2E0,V9a2_2E0,V6n2_2E0)) )
         => ( ( V8a1_2E0 = V9a2_2E0 )
            & ( V5n1_2E0 = V6n2_2E0 ) ) ) ) ).

tff(thm_2Eiterate_2EFINREC__SUC__LEMMA,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),V1b_2E0: A_27b] :
      ( ! [V2x_2E0: A_27a,V3y_2E0: A_27a,V4s_2E0: A_27b] :
          ( ( V2x_2E0 != V3y_2E0 )
         => ( app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V2x_2E0),app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V3y_2E0),V4s_2E0)) = app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V3y_2E0),app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V2x_2E0),V4s_2E0)) ) )
     => ! [V5n_2E0: tyop_2Enum_2Enum,V6s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V7z_2E0: A_27b] :
          ( p(c_2Eiterate_2EFINREC_2E5(A_27a,A_27b,V0f_2E0,V1b_2E0,V6s_2E0,V7z_2E0,c_2Enum_2ESUC_2E1(V5n_2E0)))
         => ! [V8x_2E0: A_27a] :
              ( p(c_2Ebool_2EIN_2E2(A_27a,V8x_2E0,V6s_2E0))
             => ? [V9w_2E0: A_27b] :
                  ( p(c_2Eiterate_2EFINREC_2E5(A_27a,A_27b,V0f_2E0,V1b_2E0,c_2Epred__set_2EDELETE_2E2(A_27a,V6s_2E0,V8x_2E0),V9w_2E0,V5n_2E0))
                  & ( V7z_2E0 = app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V8x_2E0),V9w_2E0) ) ) ) ) ) ).

tff(thm_2Eiterate_2EFINREC__1__LEMMA,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),V1b_2E0: A_27b,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3a_2E0: A_27b] :
      ( p(c_2Eiterate_2EFINREC_2E5(A_27a,A_27b,V0f_2E0,V1b_2E0,V2s_2E0,V3a_2E0,c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0)))
    <=> ? [V4x_2E0: A_27a] :
          ( ( V2s_2E0 = c_2Epred__set_2EINSERT_2E2(A_27a,V4x_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a)) )
          & ( V3a_2E0 = app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V0f_2E0,V4x_2E0),V1b_2E0) ) ) ) ).

tff(thm_2Eiterate_2EFINREC__def__compute,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),V1b_2E0: A_27b,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3a_2E0: A_27b] :
          ( p(c_2Eiterate_2EFINREC_2E5(A_27a,A_27b,V0f_2E0,V1b_2E0,V2s_2E0,V3a_2E0,c_2Enum_2E0_2E0))
        <=> ( ( V2s_2E0 = c_2Epred__set_2EEMPTY_2E0(A_27a) )
            & ( V3a_2E0 = V1b_2E0 ) ) )
      & ! [V4f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),V5b_2E0: A_27b,V6s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V7a_2E0: A_27b,V8n_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Eiterate_2EFINREC_2E5(A_27a,A_27b,V4f_2E0,V5b_2E0,V6s_2E0,V7a_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V8n_2E0))))
        <=> ? [V9x_2E0: A_27a,V10c_2E0: A_27b] :
              ( p(c_2Ebool_2EIN_2E2(A_27a,V9x_2E0,V6s_2E0))
              & p(c_2Eiterate_2EFINREC_2E5(A_27a,A_27b,V4f_2E0,V5b_2E0,c_2Epred__set_2EDELETE_2E2(A_27a,V6s_2E0,V9x_2E0),V10c_2E0,c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V8n_2E0)),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
              & ( V7a_2E0 = app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V4f_2E0,V9x_2E0),V10c_2E0) ) ) )
      & ! [V11f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27b)),V12b_2E0: A_27b,V13s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V14a_2E0: A_27b,V15n_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Eiterate_2EFINREC_2E5(A_27a,A_27b,V11f_2E0,V12b_2E0,V13s_2E0,V14a_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(V15n_2E0))))
        <=> ? [V16x_2E0: A_27a,V17c_2E0: A_27b] :
              ( p(c_2Ebool_2EIN_2E2(A_27a,V16x_2E0,V13s_2E0))
              & p(c_2Eiterate_2EFINREC_2E5(A_27a,A_27b,V11f_2E0,V12b_2E0,c_2Epred__set_2EDELETE_2E2(A_27a,V13s_2E0,V16x_2E0),V17c_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(V15n_2E0))))
              & ( V14a_2E0 = app_2E2(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27b),V11f_2E0,V16x_2E0),V17c_2E0) ) ) ) ) ).

tff(thm_2Eiterate_2EREAL__WLOG__LT,axiom,
    ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))] :
      ( ( ! [V1x_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V1x_2E0),V1x_2E0))
        & ! [V2x_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V2x_2E0),V3y_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V3y_2E0),V2x_2E0) )
        & ! [V4x_2E0: tyop_2Erealax_2Ereal,V5y_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Erealax_2Ereal__lt_2E2(V4x_2E0,V5y_2E0))
           => p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V4x_2E0),V5y_2E0)) ) )
     => ! [V6x_2E0: tyop_2Erealax_2Ereal,V7y_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0,V6x_2E0),V7y_2E0)) ) ).

tff(thm_2Eiterate_2EUPPER__BOUND__FINITE__SET__REAL,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
     => ? [V2a_2E0: tyop_2Erealax_2Ereal] :
        ! [V3x_2E0: A_27a] :
          ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1s_2E0))
         => p(c_2Ereal_2Ereal__lte_2E2(app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0),V2a_2E0)) ) ) ).

tff(thm_2Eiterate_2ELOWER__BOUND__FINITE__SET__REAL,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
     => ? [V2a_2E0: tyop_2Erealax_2Ereal] :
        ! [V3x_2E0: A_27a] :
          ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1s_2E0))
         => p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0))) ) ) ).

tff(thm_2Eiterate_2ESIMP__REAL__ARCH,axiom,
    ! [V0x_2E0: tyop_2Erealax_2Ereal] :
    ? [V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ereal_2Ereal__lte_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(V1n_2E0))) ).

tff(thm_2Eiterate_2EREAL__LT__BETWEEN,axiom,
    ! [V0a_2E0: tyop_2Erealax_2Ereal,V1b_2E0: tyop_2Erealax_2Ereal] :
      ( p(c_2Erealax_2Ereal__lt_2E2(V0a_2E0,V1b_2E0))
    <=> ? [V2x_2E0: tyop_2Erealax_2Ereal] :
          ( p(c_2Erealax_2Ereal__lt_2E2(V0a_2E0,V2x_2E0))
          & p(c_2Erealax_2Ereal__lt_2E2(V2x_2E0,V1b_2E0)) ) ) ).

tff(thm_2Eiterate_2EREAL__SUP__LE__EQ,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1y_2E0: tyop_2Erealax_2Ereal] :
      ( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & ? [V2b_2E0: tyop_2Erealax_2Ereal] :
          ! [V3x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2b_2E0)) ) )
     => ( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Esup_2E1(V0s_2E0),V1y_2E0))
      <=> ! [V4x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V4x_2E0,V1y_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EREAL__SUP__UNIQUE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1b_2E0: tyop_2Erealax_2Ereal] :
      ( ( ! [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V2x_2E0,V1b_2E0)) )
        & ! [V3b_27_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Erealax_2Ereal__lt_2E2(V3b_27_2E0,V1b_2E0))
           => ? [V4x_2E0: tyop_2Erealax_2Ereal] :
                ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V0s_2E0))
                & p(c_2Erealax_2Ereal__lt_2E2(V3b_27_2E0,V4x_2E0)) ) ) )
     => ( c_2Ereal_2Esup_2E1(V0s_2E0) = V1b_2E0 ) ) ).

tff(thm_2Eiterate_2EREAL__SUP__LE__S,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1b_2E0: tyop_2Erealax_2Ereal] :
      ( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & ! [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V2x_2E0,V1b_2E0)) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Esup_2E1(V0s_2E0),V1b_2E0)) ) ).

tff(thm_2Eiterate_2EREAL__SUP__LE__SUBSET,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
      ( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0))
        & ? [V2b_2E0: tyop_2Erealax_2Ereal] :
          ! [V3x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V1t_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2b_2E0)) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Esup_2E1(V0s_2E0),c_2Ereal_2Esup_2E1(V1t_2E0))) ) ).

tff(thm_2Eiterate_2EREAL__LE__SUP,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal,V3y_2E0: tyop_2Erealax_2Ereal] :
      ( ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3y_2E0,V0s_2E0))
        & p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V3y_2E0))
        & ! [V4x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V4x_2E0,V2b_2E0)) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,c_2Ereal_2Esup_2E1(V0s_2E0))) ) ).

tff(thm_2Eiterate_2EREAL__SUP__BOUNDS,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
      ( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & ! [V3x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
           => ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V3x_2E0))
              & p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2b_2E0)) ) ) )
     => ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,c_2Ereal_2Esup_2E1(V0s_2E0)))
        & p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Esup_2E1(V0s_2E0),V2b_2E0)) ) ) ).

tff(thm_2Eiterate_2EREAL__ABS__SUP__LE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
      ( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & ! [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V2x_2E0),V1a_2E0)) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Esup_2E1(V0s_2E0)),V1a_2E0)) ) ).

tff(thm_2Eiterate_2EREAL__SUP__ASCLOSE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1l_2E0: tyop_2Erealax_2Ereal,V2e_2E0: tyop_2Erealax_2Ereal] :
      ( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & ! [V3x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V3x_2E0,V1l_2E0)),V2e_2E0)) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Esup_2E1(V0s_2E0),V1l_2E0)),V2e_2E0)) ) ).

tff(thm_2Eiterate_2EREAL__INF__ASCLOSE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1l_2E0: tyop_2Erealax_2Ereal,V2e_2E0: tyop_2Erealax_2Ereal] :
      ( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & ! [V3x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V3x_2E0,V1l_2E0)),V2e_2E0)) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(c_2Eiterate_2Einf_2E1(V0s_2E0),V1l_2E0)),V2e_2E0)) ) ).

tff(thm_2Eiterate_2EREAL__ABS__INF__LE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
      ( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & ! [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V2x_2E0),V1a_2E0)) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(c_2Eiterate_2Einf_2E1(V0s_2E0)),V1a_2E0)) ) ).

tff(thm_2Eiterate_2EREAL__INF__BOUNDS,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal,V2b_2E0: tyop_2Erealax_2Ereal] :
      ( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & ! [V3x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
           => ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V3x_2E0))
              & p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2b_2E0)) ) ) )
     => ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,c_2Eiterate_2Einf_2E1(V0s_2E0)))
        & p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2Einf_2E1(V0s_2E0),V2b_2E0)) ) ) ).

tff(thm_2Eiterate_2EREAL__LE__INF__SUBSET,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
      ( ( ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & p(c_2Epred__set_2ESUBSET_2E2(tyop_2Erealax_2Ereal,V1t_2E0,V0s_2E0))
        & ? [V2b_2E0: tyop_2Erealax_2Ereal] :
          ! [V3x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V2b_2E0,V3x_2E0)) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2Einf_2E1(V0s_2E0),c_2Eiterate_2Einf_2E1(V1t_2E0))) ) ).

tff(thm_2Eiterate_2EREAL__LE__INF,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1b_2E0: tyop_2Erealax_2Ereal] :
      ( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & ! [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V1b_2E0,V2x_2E0)) ) )
     => p(c_2Ereal_2Ereal__lte_2E2(V1b_2E0,c_2Eiterate_2Einf_2E1(V0s_2E0))) ) ).

tff(thm_2Eiterate_2EREAL__INF__UNIQUE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1b_2E0: tyop_2Erealax_2Ereal] :
      ( ( ! [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V1b_2E0,V2x_2E0)) )
        & ! [V3b_27_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Erealax_2Ereal__lt_2E2(V1b_2E0,V3b_27_2E0))
           => ? [V4x_2E0: tyop_2Erealax_2Ereal] :
                ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V0s_2E0))
                & p(c_2Erealax_2Ereal__lt_2E2(V4x_2E0,V3b_27_2E0)) ) ) )
     => ( c_2Eiterate_2Einf_2E1(V0s_2E0) = V1b_2E0 ) ) ).

tff(thm_2Eiterate_2EREAL__INF__LT__FINITE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ( p(c_2Erealax_2Ereal__lt_2E2(c_2Eiterate_2Einf_2E1(V0s_2E0),V1a_2E0))
      <=> ? [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
            & p(c_2Erealax_2Ereal__lt_2E2(V2x_2E0,V1a_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EREAL__LT__INF__FINITE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ( p(c_2Erealax_2Ereal__lt_2E2(V1a_2E0,c_2Eiterate_2Einf_2E1(V0s_2E0)))
      <=> ! [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
           => p(c_2Erealax_2Ereal__lt_2E2(V1a_2E0,V2x_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EREAL__INF__LE__FINITE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ( p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2Einf_2E1(V0s_2E0),V1a_2E0))
      <=> ? [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
            & p(c_2Ereal_2Ereal__lte_2E2(V2x_2E0,V1a_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EREAL__LE__INF__FINITE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1a_2E0: tyop_2Erealax_2Ereal] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ( p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,c_2Eiterate_2Einf_2E1(V0s_2E0)))
      <=> ! [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V1a_2E0,V2x_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EINF__FINITE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,c_2Eiterate_2Einf_2E1(V0s_2E0),V0s_2E0))
        & ! [V1x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2Einf_2E1(V0s_2E0),V1x_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EINF__FINITE__LEMMA,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ? [V1b_2E0: tyop_2Erealax_2Ereal] :
          ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V1b_2E0,V0s_2E0))
          & ! [V2x_2E0: tyop_2Erealax_2Ereal] :
              ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
             => p(c_2Ereal_2Ereal__lte_2E2(V1b_2E0,V2x_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EINF,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
      ( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & ? [V1b_2E0: tyop_2Erealax_2Ereal] :
          ! [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V1b_2E0,V2x_2E0)) ) )
     => ( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Eiterate_2Einf_2E1(V0s_2E0),V3x_2E0)) )
        & ! [V4b_2E0: tyop_2Erealax_2Ereal] :
            ( ! [V5x_2E0: tyop_2Erealax_2Ereal] :
                ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V0s_2E0))
               => p(c_2Ereal_2Ereal__lte_2E2(V4b_2E0,V5x_2E0)) )
           => p(c_2Ereal_2Ereal__lte_2E2(V4b_2E0,c_2Eiterate_2Einf_2E1(V0s_2E0))) ) ) ) ).

tff(thm_2Eiterate_2EINF__EQ,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
      ( ! [V2a_2E0: tyop_2Erealax_2Ereal] :
          ( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
              ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
             => p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V3x_2E0)) )
        <=> ! [V4x_2E0: tyop_2Erealax_2Ereal] :
              ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V4x_2E0,V1t_2E0))
             => p(c_2Ereal_2Ereal__lte_2E2(V2a_2E0,V4x_2E0)) ) )
     => ( c_2Eiterate_2Einf_2E1(V0s_2E0) = c_2Eiterate_2Einf_2E1(V1t_2E0) ) ) ).

tff(thm_2Eiterate_2ESUP__UNION,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
      ( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & ( V1t_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & ? [V2b_2E0: tyop_2Erealax_2Ereal] :
          ! [V3x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2b_2E0)) )
        & ? [V4c_2E0: tyop_2Erealax_2Ereal] :
          ! [V5x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V5x_2E0,V1t_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(V5x_2E0,V4c_2E0)) ) )
     => ( c_2Ereal_2Esup_2E1(c_2Epred__set_2EUNION_2E2(tyop_2Erealax_2Ereal,V0s_2E0,V1t_2E0)) = c_2Ereal_2Emax_2E2(c_2Ereal_2Esup_2E1(V0s_2E0),c_2Ereal_2Esup_2E1(V1t_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUP__UNIQUE,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V1b_2E0: tyop_2Erealax_2Ereal] :
      ( ! [V2c_2E0: tyop_2Erealax_2Ereal] :
          ( ! [V3x_2E0: tyop_2Erealax_2Ereal] :
              ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V3x_2E0,V0s_2E0))
             => p(c_2Ereal_2Ereal__lte_2E2(V3x_2E0,V2c_2E0)) )
        <=> p(c_2Ereal_2Ereal__lte_2E2(V1b_2E0,V2c_2E0)) )
     => ( c_2Ereal_2Esup_2E1(V0s_2E0) = V1b_2E0 ) ) ).

tff(thm_2Eiterate_2ESUP__SING,axiom,
    ! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Esup_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) = V0a_2E0 ) ).

tff(thm_2Eiterate_2ESUP__INSERT__FINITE,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
      ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
          ( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V1s_2E0))
        <=> ( V1s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1s_2E0))
         => ( c_2Ereal_2Esup_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V1s_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V1s_2E0),V0x_2E0,c_2Ereal_2Emax_2E2(V0x_2E0,c_2Ereal_2Esup_2E1(V1s_2E0))) ) ) ) ).

tff(thm_2Eiterate_2EINF__INSERT__FINITE,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
      ( ! [V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
          ( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V1s_2E0))
        <=> ( V1s_2E0 = c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) ) )
     => ! [V0x_2E0: tyop_2Erealax_2Ereal,V1s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,V1s_2E0))
         => ( c_2Eiterate_2Einf_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0x_2E0,V1s_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,F0_2E0,V1s_2E0),V0x_2E0,c_2Ereal_2Emin_2E2(V0x_2E0,c_2Eiterate_2Einf_2E1(V1s_2E0))) ) ) ) ).

tff(thm_2Eiterate_2EREAL__SUP__EQ__INF,axiom,
    ! [V0s_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] :
      ( ( ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal) )
        & ? [V1B_2E0: tyop_2Erealax_2Ereal] :
          ! [V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Ebool_2EIN_2E2(tyop_2Erealax_2Ereal,V2x_2E0,V0s_2E0))
           => p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Eabs_2E1(V2x_2E0),V1B_2E0)) ) )
     => ( ( c_2Ereal_2Esup_2E1(V0s_2E0) = c_2Eiterate_2Einf_2E1(V0s_2E0) )
      <=> ? [V3a_2E0: tyop_2Erealax_2Ereal] : ( V0s_2E0 = c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V3a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal)) ) ) ) ).

tff(thm_2Eiterate_2EINF__SING,axiom,
    ! [V0a_2E0: tyop_2Erealax_2Ereal] : ( c_2Eiterate_2Einf_2E1(c_2Epred__set_2EINSERT_2E2(tyop_2Erealax_2Ereal,V0a_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Erealax_2Ereal))) = V0a_2E0 ) ).

tff(thm_2Eiterate_2EFINITE__NUMSEG,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1n_2E0))) ).

tff(thm_2Eiterate_2ENUMSEG__COMBINE__R,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1p_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,c_2Earithmetic_2E_2B_2E2(V1p_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
        & p(c_2Earithmetic_2E_3C_3D_2E2(V1p_2E0,V2n_2E0)) )
     => ( c_2Epred__set_2EUNION_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1p_2E0),c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2B_2E2(V1p_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V2n_2E0)) = c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V2n_2E0) ) ) ).

tff(thm_2Eiterate_2ENUMSEG__COMBINE__L,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1p_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1p_2E0))
        & p(c_2Earithmetic_2E_3C_3D_2E2(V1p_2E0,c_2Earithmetic_2E_2B_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
     => ( c_2Epred__set_2EUNION_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,c_2Earithmetic_2E_2D_2E2(V1p_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Eiterate_2E_2E_2E_2E2(V1p_2E0,V2n_2E0)) = c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V2n_2E0) ) ) ).

tff(thm_2Eiterate_2ENUMSEG__LREC,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
     => ( c_2Epred__set_2EINSERT_2E2(tyop_2Enum_2Enum,V0m_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V1n_2E0)) = c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1n_2E0) ) ) ).

tff(thm_2Eiterate_2ENUMSEG__RREC,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0))
     => ( c_2Epred__set_2EINSERT_2E2(tyop_2Enum_2Enum,V1n_2E0,c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,c_2Earithmetic_2E_2D_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) = c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1n_2E0) ) ) ).

tff(thm_2Eiterate_2ENUMSEG__REC,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)))
     => ( c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)) = c_2Epred__set_2EINSERT_2E2(tyop_2Enum_2Enum,c_2Enum_2ESUC_2E1(V1n_2E0),c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1n_2E0)) ) ) ).

tff(thm_2Eiterate_2EIN__NUMSEG,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V2p_2E0,c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1n_2E0)))
    <=> ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V2p_2E0))
        & p(c_2Earithmetic_2E_3C_3D_2E2(V2p_2E0,V1n_2E0)) ) ) ).

tff(thm_2Eiterate_2EIN__NUMSEG__0,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V0m_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V1n_2E0)) = c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,V1n_2E0) ) ).

tff(thm_2Eiterate_2ENUMSEG__SING,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2E_2E_2E_2E2(V0n_2E0,V0n_2E0) = c_2Epred__set_2EINSERT_2E2(tyop_2Enum_2Enum,V0n_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Enum_2Enum)) ) ).

tff(thm_2Eiterate_2ENUMSEG__EMPTY,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1n_2E0) = c_2Epred__set_2EEMPTY_2E0(tyop_2Enum_2Enum) )
    <=> p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V0m_2E0)) ) ).

tff(thm_2Eiterate_2ECARD__NUMSEG__LEMMA,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1d_2E0: tyop_2Enum_2Enum] : ( c_2Epred__set_2ECARD_2E1(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,c_2Earithmetic_2E_2B_2E2(V0m_2E0,V1d_2E0))) = c_2Earithmetic_2E_2B_2E2(V1d_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))) ) ).

tff(thm_2Eiterate_2ECARD__NUMSEG,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Epred__set_2ECARD_2E1(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1n_2E0)) = c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2B_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0m_2E0) ) ).

tff(thm_2Eiterate_2EHAS__SIZE__NUMSEG,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : p(c_2Ecardinal_2EHAS__SIZE_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1n_2E0),c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2B_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0m_2E0))) ).

tff(thm_2Eiterate_2ECARD__NUMSEG__1,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Epred__set_2ECARD_2E1(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0)) = V0n_2E0 ) ).

tff(thm_2Eiterate_2EHAS__SIZE__NUMSEG__1,axiom,
    ! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Ecardinal_2EHAS__SIZE_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0),V0n_2E0)) ).

tff(thm_2Eiterate_2ENUMSEG__CLAUSES,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( ! [V0m_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V0m_2E0))
        <=> ( V0m_2E0 = c_2Enum_2E0_2E0 ) )
     => ( ! [V0m_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,c_2Enum_2E0_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V0m_2E0),c_2Epred__set_2EINSERT_2E2(tyop_2Enum_2Enum,c_2Enum_2E0_2E0,c_2Epred__set_2EEMPTY_2E0(tyop_2Enum_2Enum)),c_2Epred__set_2EEMPTY_2E0(tyop_2Enum_2Enum)) )
        & ! [V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)),c_2Epred__set_2EINSERT_2E2(tyop_2Enum_2Enum,c_2Enum_2ESUC_2E1(V2n_2E0),c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0)),c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EFINITE__INDEX__NUMSEG,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
    <=> ? [V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] :
          ( ! [V2i_2E0: tyop_2Enum_2Enum,V3j_2E0: tyop_2Enum_2Enum] :
              ( ( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V2i_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0))))
                & p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V3j_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0))))
                & ( app_2E2(tyop_2Enum_2Enum,A_27a,V1f_2E0,V2i_2E0) = app_2E2(tyop_2Enum_2Enum,A_27a,V1f_2E0,V3j_2E0) ) )
             => ( V2i_2E0 = V3j_2E0 ) )
          & ( V0s_2E0 = c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,A_27a,V1f_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0))) ) ) ) ).

tff(thm_2Eiterate_2EFINITE__INDEX__NUMBERS,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
    <=> ? [V1k_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] :
          ( ! [V3i_2E0: tyop_2Enum_2Enum,V4j_2E0: tyop_2Enum_2Enum] :
              ( ( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V3i_2E0,V1k_2E0))
                & p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4j_2E0,V1k_2E0))
                & ( app_2E2(tyop_2Enum_2Enum,A_27a,V2f_2E0,V3i_2E0) = app_2E2(tyop_2Enum_2Enum,A_27a,V2f_2E0,V4j_2E0) ) )
             => ( V3i_2E0 = V4j_2E0 ) )
          & p(c_2Epred__set_2EFINITE_2E1(tyop_2Enum_2Enum,V1k_2E0))
          & ( V0s_2E0 = c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,A_27a,V2f_2E0,V1k_2E0) ) ) ) ).

tff(thm_2Eiterate_2EDISJOINT__NUMSEG,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum,V3q_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epred__set_2EDISJOINT_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1n_2E0),c_2Eiterate_2E_2E_2E_2E2(V2p_2E0,V3q_2E0)))
    <=> ( p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V2p_2E0))
        | p(c_2Eprim__rec_2E_3C_2E2(V3q_2E0,V0m_2E0))
        | p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V0m_2E0))
        | p(c_2Eprim__rec_2E_3C_2E2(V3q_2E0,V2p_2E0)) ) ) ).

tff(thm_2Eiterate_2ENUMSEG__ADD__SPLIT,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Earithmetic_2E_3C_3D_2E2(V0m_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
     => ( c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0)) = c_2Epred__set_2EUNION_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1n_2E0),c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2B_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0))) ) ) ).

tff(thm_2Eiterate_2ENUMSEG__OFFSET__IMAGE,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))] :
      ( ! [V2p_2E0: tyop_2Enum_2Enum,V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F0_2E0,V2p_2E0),V3i_2E0) = c_2Earithmetic_2E_2B_2E2(V3i_2E0,V2p_2E0) )
     => ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2B_2E2(V0m_2E0,V2p_2E0),c_2Earithmetic_2E_2B_2E2(V1n_2E0,V2p_2E0)) = c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F0_2E0,V2p_2E0),c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1n_2E0)) ) ) ).

tff(thm_2Eiterate_2ESUBSET__NUMSEG,axiom,
    ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum,V2p_2E0: tyop_2Enum_2Enum,V3q_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Epred__set_2ESUBSET_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V0m_2E0,V1n_2E0),c_2Eiterate_2E_2E_2E_2E2(V2p_2E0,V3q_2E0)))
    <=> ( p(c_2Eprim__rec_2E_3C_2E2(V1n_2E0,V0m_2E0))
        | ( p(c_2Earithmetic_2E_3C_3D_2E2(V2p_2E0,V0m_2E0))
          & p(c_2Earithmetic_2E_3C_3D_2E2(V1n_2E0,V3q_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ENUMSEG__LE,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))] :
      ( ! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0,V0n_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1x_2E0,c_2Earithmetic_2E_3C_3D_2E2(V1x_2E0,V0n_2E0)) )
     => ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0,V0n_2E0)) = c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V0n_2E0) ) ) ).

tff(thm_2Eiterate_2ENUMSEG__LT,axiom,
    ! [F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))] :
      ( ! [V0n_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F1_2E0,V0n_2E0))
        <=> ( V0n_2E0 = c_2Enum_2E0_2E0 ) )
     => ( ! [V0n_2E0: tyop_2Enum_2Enum,V1x_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0,V0n_2E0),V1x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V1x_2E0,c_2Eprim__rec_2E_3C_2E2(V1x_2E0,V0n_2E0)) )
       => ! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0,V0n_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F1_2E0,V0n_2E0),c_2Epred__set_2EEMPTY_2E0(tyop_2Enum_2Enum),c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2E_2D_2E2(V0n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) ) ) ) ).

tff(thm_2Eiterate_2ETOPOLOGICAL__SORT,axiom,
    ! [A_27a: $tType,V0_3C_3C_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
      ( ( ! [V1x_2E0: A_27a,V2y_2E0: A_27a] :
            ( ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0_3C_3C_2E0,V1x_2E0),V2y_2E0))
              & p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0_3C_3C_2E0,V2y_2E0),V1x_2E0)) )
           => ( V1x_2E0 = V2y_2E0 ) )
        & ! [V3x_2E0: A_27a,V4y_2E0: A_27a,V5z_2E0: A_27a] :
            ( ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0_3C_3C_2E0,V3x_2E0),V4y_2E0))
              & p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0_3C_3C_2E0,V4y_2E0),V5z_2E0)) )
           => p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0_3C_3C_2E0,V3x_2E0),V5z_2E0)) ) )
     => ! [V6n_2E0: tyop_2Enum_2Enum,V7s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(c_2Ecardinal_2EHAS__SIZE_2E2(A_27a,V7s_2E0,V6n_2E0))
         => ? [V8f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] :
              ( ( V7s_2E0 = c_2Epred__set_2EIMAGE_2E2(tyop_2Enum_2Enum,A_27a,V8f_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V6n_2E0)) )
              & ! [V9j_2E0: tyop_2Enum_2Enum,V10k_2E0: tyop_2Enum_2Enum] :
                  ( ( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V9j_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V6n_2E0)))
                    & p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V10k_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V6n_2E0)))
                    & p(c_2Eprim__rec_2E_3C_2E2(V9j_2E0,V10k_2E0)) )
                 => ~ p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0_3C_3C_2E0,app_2E2(tyop_2Enum_2Enum,A_27a,V8f_2E0,V10k_2E0)),app_2E2(tyop_2Enum_2Enum,A_27a,V8f_2E0,V9j_2E0))) ) ) ) ) ).

tff(thm_2Eiterate_2EMONOIDAL__AC,axiom,
    ! [A_27a: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27a,V0op_2E0))
     => ( ! [V1a_2E0: A_27a] : ( app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,c_2Eiterate_2Eneutral_2E1(A_27a,V0op_2E0)),V1a_2E0) = V1a_2E0 )
        & ! [V2a_2E0: A_27a] : ( app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V2a_2E0),c_2Eiterate_2Eneutral_2E1(A_27a,V0op_2E0)) = V2a_2E0 )
        & ! [V3a_2E0: A_27a,V4b_2E0: A_27a] : ( app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V3a_2E0),V4b_2E0) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V4b_2E0),V3a_2E0) )
        & ! [V5a_2E0: A_27a,V6b_2E0: A_27a,V7c_2E0: A_27a] : ( app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V5a_2E0),V6b_2E0)),V7c_2E0) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V5a_2E0),app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V6b_2E0),V7c_2E0)) )
        & ! [V8a_2E0: A_27a,V9b_2E0: A_27a,V10c_2E0: A_27a] : ( app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V8a_2E0),app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V9b_2E0),V10c_2E0)) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V9b_2E0),app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,V8a_2E0),V10c_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EIN__SUPPORT,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V2x_2E0: A_27b,V3s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
      ( p(c_2Ebool_2EIN_2E2(A_27b,V2x_2E0,c_2Eiterate_2Esupport_2E3(A_27b,A_27a,V0op_2E0,V1f_2E0,V3s_2E0)))
    <=> ( p(c_2Ebool_2EIN_2E2(A_27b,V2x_2E0,V3s_2E0))
        & ( app_2E2(A_27b,A_27a,V1f_2E0,V2x_2E0) != c_2Eiterate_2Eneutral_2E1(A_27a,V0op_2E0) ) ) ) ).

tff(thm_2Eiterate_2ESUPPORT__SUPPORT,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27b,A_27a,V0op_2E0,V1f_2E0,c_2Eiterate_2Esupport_2E3(A_27b,A_27a,V0op_2E0,V1f_2E0,V2s_2E0)) = c_2Eiterate_2Esupport_2E3(A_27b,A_27a,V0op_2E0,V1f_2E0,V2s_2E0) ) ).

tff(thm_2Eiterate_2ESUPPORT__EMPTY,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
      ( ! [V3x_2E0: A_27b] :
          ( p(c_2Ebool_2EIN_2E2(A_27b,V3x_2E0,V2s_2E0))
         => ( app_2E2(A_27b,A_27a,V1f_2E0,V3x_2E0) = c_2Eiterate_2Eneutral_2E1(A_27a,V0op_2E0) ) )
    <=> ( c_2Eiterate_2Esupport_2E3(A_27b,A_27a,V0op_2E0,V1f_2E0,V2s_2E0) = c_2Epred__set_2EEMPTY_2E0(A_27b) ) ) ).

tff(thm_2Eiterate_2ESUPPORT__SUBSET,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : p(c_2Epred__set_2ESUBSET_2E2(A_27b,c_2Eiterate_2Esupport_2E3(A_27b,A_27a,V0op_2E0,V1f_2E0,V2s_2E0),V2s_2E0)) ).

tff(thm_2Eiterate_2EFINITE__SUPPORT,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
      ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V2s_2E0))
     => p(c_2Epred__set_2EFINITE_2E1(A_27b,c_2Eiterate_2Esupport_2E3(A_27b,A_27a,V0op_2E0,V1f_2E0,V2s_2E0))) ) ).

tff(thm_2Eiterate_2ESUPPORT__CLAUSES,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,A_27h: $tType,A_27i: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool)))] :
      ( ! [V2f_2E0: tyop_2Emin_2Efun(A_27c,A_27b),V3x_2E0: A_27c,V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
          ( p(app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool,app_2E2(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool)),F0_2E0,V2f_2E0),V3x_2E0),V0op_2E0))
        <=> ( app_2E2(A_27c,A_27b,V2f_2E0,V3x_2E0) = c_2Eiterate_2Eneutral_2E1(A_27b,V0op_2E0) ) )
     => ! [V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
          ( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] : ( c_2Eiterate_2Esupport_2E3(A_27a,A_27b,V0op_2E0,V1f_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a)) = c_2Epred__set_2EEMPTY_2E0(A_27a) )
          & ! [V2f_2E0: tyop_2Emin_2Efun(A_27c,A_27b),V3x_2E0: A_27c,V4s_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27c,A_27b,V0op_2E0,V2f_2E0,c_2Epred__set_2EINSERT_2E2(A_27c,V3x_2E0,V4s_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool,app_2E2(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Ebool)),F0_2E0,V2f_2E0),V3x_2E0),V0op_2E0),c_2Eiterate_2Esupport_2E3(A_27c,A_27b,V0op_2E0,V2f_2E0,V4s_2E0),c_2Epred__set_2EINSERT_2E2(A_27c,V3x_2E0,c_2Eiterate_2Esupport_2E3(A_27c,A_27b,V0op_2E0,V2f_2E0,V4s_2E0))) )
          & ! [V5f_2E0: tyop_2Emin_2Efun(A_27d,A_27b),V6x_2E0: A_27d,V7s_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27d,A_27b,V0op_2E0,V5f_2E0,c_2Epred__set_2EDELETE_2E2(A_27d,V7s_2E0,V6x_2E0)) = c_2Epred__set_2EDELETE_2E2(A_27d,c_2Eiterate_2Esupport_2E3(A_27d,A_27b,V0op_2E0,V5f_2E0,V7s_2E0),V6x_2E0) )
          & ! [V8f_2E0: tyop_2Emin_2Efun(A_27e,A_27b),V9s_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),V10t_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27e,A_27b,V0op_2E0,V8f_2E0,c_2Epred__set_2EUNION_2E2(A_27e,V9s_2E0,V10t_2E0)) = c_2Epred__set_2EUNION_2E2(A_27e,c_2Eiterate_2Esupport_2E3(A_27e,A_27b,V0op_2E0,V8f_2E0,V9s_2E0),c_2Eiterate_2Esupport_2E3(A_27e,A_27b,V0op_2E0,V8f_2E0,V10t_2E0)) )
          & ! [V11f_2E0: tyop_2Emin_2Efun(A_27f,A_27b),V12s_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool),V13t_2E0: tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27f,A_27b,V0op_2E0,V11f_2E0,c_2Epred__set_2EINTER_2E2(A_27f,V12s_2E0,V13t_2E0)) = c_2Epred__set_2EINTER_2E2(A_27f,c_2Eiterate_2Esupport_2E3(A_27f,A_27b,V0op_2E0,V11f_2E0,V12s_2E0),c_2Eiterate_2Esupport_2E3(A_27f,A_27b,V0op_2E0,V11f_2E0,V13t_2E0)) )
          & ! [V14f_2E0: tyop_2Emin_2Efun(A_27g,A_27b),V15s_2E0: tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),V16t_2E0: tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27g,A_27b,V0op_2E0,V14f_2E0,c_2Epred__set_2EDIFF_2E2(A_27g,V15s_2E0,V16t_2E0)) = c_2Epred__set_2EDIFF_2E2(A_27g,c_2Eiterate_2Esupport_2E3(A_27g,A_27b,V0op_2E0,V14f_2E0,V15s_2E0),c_2Eiterate_2Esupport_2E3(A_27g,A_27b,V0op_2E0,V14f_2E0,V16t_2E0)) )
          & ! [V17f_2E0: tyop_2Emin_2Efun(A_27h,A_27i),V18g_2E0: tyop_2Emin_2Efun(A_27i,A_27b),V19s_2E0: tyop_2Emin_2Efun(A_27h,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Esupport_2E3(A_27i,A_27b,V0op_2E0,V18g_2E0,c_2Epred__set_2EIMAGE_2E2(A_27h,A_27i,V17f_2E0,V19s_2E0)) = c_2Epred__set_2EIMAGE_2E2(A_27h,A_27i,V17f_2E0,c_2Eiterate_2Esupport_2E3(A_27h,A_27b,V0op_2E0,c_2Ecombin_2Eo_2E2(A_27h,A_27b,A_27i,V18g_2E0,V17f_2E0),V19s_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ESUPPORT__DELTA,axiom,
    ! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a))))] :
      ( ! [V4x_2E0: A_27b,V3a_2E0: A_27b] :
          ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),F1_2E0,V4x_2E0),V3a_2E0))
        <=> ( V4x_2E0 = V3a_2E0 ) )
     => ( ! [V3a_2E0: A_27b,V2f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),V4x_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a))),F0_2E0,V3a_2E0),V2f_2E0),V0op_2E0),V4x_2E0) = c_2Ebool_2ECOND_2E3(A_27a,app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),F1_2E0,V4x_2E0),V3a_2E0),app_2E2(A_27b,A_27a,V2f_2E0,V4x_2E0),c_2Eiterate_2Eneutral_2E1(A_27a,V0op_2E0)) )
       => ! [V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),V1s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V3a_2E0: A_27b] : ( c_2Eiterate_2Esupport_2E3(A_27b,A_27a,V0op_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a))),F0_2E0,V3a_2E0),V2f_2E0),V0op_2E0),V1s_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E2(A_27b,V3a_2E0,V1s_2E0),c_2Eiterate_2Esupport_2E3(A_27b,A_27a,V0op_2E0,V2f_2E0,c_2Epred__set_2EINSERT_2E2(A_27b,V3a_2E0,c_2Epred__set_2EEMPTY_2E0(A_27b))),c_2Epred__set_2EEMPTY_2E0(A_27b)) ) ) ) ).

tff(thm_2Eiterate_2EFINITE__SUPPORT__DELTA,axiom,
    ! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a))))] :
      ( ! [V4x_2E0: A_27b,V3a_2E0: A_27b] :
          ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),F1_2E0,V4x_2E0),V3a_2E0))
        <=> ( V4x_2E0 = V3a_2E0 ) )
     => ( ! [V3a_2E0: A_27b,V2f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V1op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),V4x_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a))),F0_2E0,V3a_2E0),V2f_2E0),V1op_2E0),V4x_2E0) = c_2Ebool_2ECOND_2E3(A_27a,app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),F1_2E0,V4x_2E0),V3a_2E0),app_2E2(A_27b,A_27a,V2f_2E0,V4x_2E0),c_2Eiterate_2Eneutral_2E1(A_27a,V1op_2E0)) )
       => ! [V0s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),V2f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V3a_2E0: A_27b] : p(c_2Epred__set_2EFINITE_2E1(A_27b,c_2Eiterate_2Esupport_2E3(A_27b,A_27a,V1op_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a))),F0_2E0,V3a_2E0),V2f_2E0),V1op_2E0),V0s_2E0))) ) ) ).

tff(thm_2Eiterate_2EITERATE__SUPPORT,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,c_2Eiterate_2Esupport_2E3(A_27b,A_27a,V0op_2E0,V1f_2E0,V2s_2E0),V1f_2E0) = c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,V2s_2E0,V1f_2E0) ) ).

tff(thm_2Eiterate_2EITERATE__EXPAND__CASES,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,V2s_2E0,V1f_2E0) = c_2Ebool_2ECOND_2E3(A_27a,c_2Epred__set_2EFINITE_2E1(A_27b,c_2Eiterate_2Esupport_2E3(A_27b,A_27a,V0op_2E0,V1f_2E0,V2s_2E0)),c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,c_2Eiterate_2Esupport_2E3(A_27b,A_27a,V0op_2E0,V1f_2E0,V2s_2E0),V1f_2E0),c_2Eiterate_2Eneutral_2E1(A_27a,V0op_2E0)) ) ).

tff(thm_2Eiterate_2EITERATE__CLAUSES__GEN,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
     => ( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] : ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a),V1f_2E0) = c_2Eiterate_2Eneutral_2E1(A_27b,V0op_2E0) )
        & ! [V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V3x_2E0: A_27a,V4s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
            ( ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
              & p(c_2Epred__set_2EFINITE_2E1(A_27a,c_2Eiterate_2Esupport_2E3(A_27a,A_27b,V0op_2E0,V2f_2E0,V4s_2E0))) )
           => ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,c_2Epred__set_2EINSERT_2E2(A_27a,V3x_2E0,V4s_2E0),V2f_2E0) = c_2Ebool_2ECOND_2E3(A_27b,c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V4s_2E0),c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V4s_2E0,V2f_2E0),app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),V0op_2E0,app_2E2(A_27a,A_27b,V2f_2E0,V3x_2E0)),c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V4s_2E0,V2f_2E0))) ) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__CLAUSES,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27a,V0op_2E0))
     => ( ! [V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a)] : ( c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,c_2Epred__set_2EEMPTY_2E0(A_27b),V1f_2E0) = c_2Eiterate_2Eneutral_2E1(A_27a,V0op_2E0) )
        & ! [V2f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V3x_2E0: A_27b,V4s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
            ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V4s_2E0))
           => ( c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,c_2Epred__set_2EINSERT_2E2(A_27b,V3x_2E0,V4s_2E0),V2f_2E0) = c_2Ebool_2ECOND_2E3(A_27a,c_2Ebool_2EIN_2E2(A_27b,V3x_2E0,V4s_2E0),c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,V4s_2E0,V2f_2E0),app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,app_2E2(A_27b,A_27a,V2f_2E0,V3x_2E0)),c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,V4s_2E0,V2f_2E0))) ) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__UNION,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27a,V0op_2E0))
     => ! [V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
          ( ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V2s_2E0))
            & p(c_2Epred__set_2EFINITE_2E1(A_27b,V3t_2E0))
            & p(c_2Epred__set_2EDISJOINT_2E2(A_27b,V2s_2E0,V3t_2E0)) )
         => ( c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,c_2Epred__set_2EUNION_2E2(A_27b,V2s_2E0,V3t_2E0),V1f_2E0) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,V2s_2E0,V1f_2E0)),c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,V3t_2E0,V1f_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__UNION__GEN,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
     => ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,c_2Eiterate_2Esupport_2E3(A_27a,A_27b,V0op_2E0,V1f_2E0,V2s_2E0)))
            & p(c_2Epred__set_2EFINITE_2E1(A_27a,c_2Eiterate_2Esupport_2E3(A_27a,A_27b,V0op_2E0,V1f_2E0,V3t_2E0)))
            & p(c_2Epred__set_2EDISJOINT_2E2(A_27a,c_2Eiterate_2Esupport_2E3(A_27a,A_27b,V0op_2E0,V1f_2E0,V2s_2E0),c_2Eiterate_2Esupport_2E3(A_27a,A_27b,V0op_2E0,V1f_2E0,V3t_2E0))) )
         => ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,c_2Epred__set_2EUNION_2E2(A_27a,V2s_2E0,V3t_2E0),V1f_2E0) = app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),V0op_2E0,c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V2s_2E0,V1f_2E0)),c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V3t_2E0,V1f_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__DIFF,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27a,V0op_2E0))
     => ! [V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
          ( ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V2s_2E0))
            & p(c_2Epred__set_2ESUBSET_2E2(A_27b,V3t_2E0,V2s_2E0)) )
         => ( app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,c_2Epred__set_2EDIFF_2E2(A_27b,V2s_2E0,V3t_2E0),V1f_2E0)),c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,V3t_2E0,V1f_2E0)) = c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,V2s_2E0,V1f_2E0) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__DIFF__GEN,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
     => ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,c_2Eiterate_2Esupport_2E3(A_27a,A_27b,V0op_2E0,V1f_2E0,V2s_2E0)))
            & p(c_2Epred__set_2ESUBSET_2E2(A_27a,c_2Eiterate_2Esupport_2E3(A_27a,A_27b,V0op_2E0,V1f_2E0,V3t_2E0),c_2Eiterate_2Esupport_2E3(A_27a,A_27b,V0op_2E0,V1f_2E0,V2s_2E0))) )
         => ( app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),V0op_2E0,c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,c_2Epred__set_2EDIFF_2E2(A_27a,V2s_2E0,V3t_2E0),V1f_2E0)),c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V3t_2E0,V1f_2E0)) = c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V2s_2E0,V1f_2E0) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__INCL__EXCL,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27a,V0op_2E0))
     => ! [V1s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3f_2E0: tyop_2Emin_2Efun(A_27b,A_27a)] :
          ( ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V1s_2E0))
            & p(c_2Epred__set_2EFINITE_2E1(A_27b,V2t_2E0)) )
         => ( app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,V1s_2E0,V3f_2E0)),c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,V2t_2E0,V3f_2E0)) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,c_2Epred__set_2EUNION_2E2(A_27b,V1s_2E0,V2t_2E0),V3f_2E0)),c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,c_2Epred__set_2EINTER_2E2(A_27b,V1s_2E0,V2t_2E0),V3f_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__CLOSED,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
     => ! [V1P_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
          ( ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,c_2Eiterate_2Eneutral_2E1(A_27b,V0op_2E0)))
            & ! [V2x_2E0: A_27b,V3y_2E0: A_27b] :
                ( ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,V2x_2E0))
                  & p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,V3y_2E0)) )
               => p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),V0op_2E0,V2x_2E0),V3y_2E0))) ) )
         => ! [V4f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V5s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
              ( ! [V6x_2E0: A_27a] :
                  ( ( p(c_2Ebool_2EIN_2E2(A_27a,V6x_2E0,V5s_2E0))
                    & ( app_2E2(A_27a,A_27b,V4f_2E0,V6x_2E0) != c_2Eiterate_2Eneutral_2E1(A_27b,V0op_2E0) ) )
                 => p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,app_2E2(A_27a,A_27b,V4f_2E0,V6x_2E0))) )
             => p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1P_2E0,c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V5s_2E0,V4f_2E0))) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__RELATED,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
     => ! [V1R_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
          ( ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1R_2E0,c_2Eiterate_2Eneutral_2E1(A_27b,V0op_2E0)),c_2Eiterate_2Eneutral_2E1(A_27b,V0op_2E0)))
            & ! [V2x1_2E0: A_27b,V3y1_2E0: A_27b,V4x2_2E0: A_27b,V5y2_2E0: A_27b] :
                ( ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1R_2E0,V2x1_2E0),V4x2_2E0))
                  & p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1R_2E0,V3y1_2E0),V5y2_2E0)) )
               => p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1R_2E0,app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),V0op_2E0,V2x1_2E0),V3y1_2E0)),app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),V0op_2E0,V4x2_2E0),V5y2_2E0))) ) )
         => ! [V6f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V7g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V8s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
              ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V8s_2E0))
                & ! [V9x_2E0: A_27a] :
                    ( p(c_2Ebool_2EIN_2E2(A_27a,V9x_2E0,V8s_2E0))
                   => p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1R_2E0,app_2E2(A_27a,A_27b,V6f_2E0,V9x_2E0)),app_2E2(A_27a,A_27b,V7g_2E0,V9x_2E0))) ) )
             => p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1R_2E0,c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V8s_2E0,V6f_2E0)),c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V8s_2E0,V7g_2E0))) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__EQ__NEUTRAL,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
     => ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( ! [V3x_2E0: A_27a] :
              ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
             => ( app_2E2(A_27a,A_27b,V1f_2E0,V3x_2E0) = c_2Eiterate_2Eneutral_2E1(A_27b,V0op_2E0) ) )
         => ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V2s_2E0,V1f_2E0) = c_2Eiterate_2Eneutral_2E1(A_27b,V0op_2E0) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__SING,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
     => ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2x_2E0: A_27a] : ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,c_2Epred__set_2EINSERT_2E2(A_27a,V2x_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a)),V1f_2E0) = app_2E2(A_27a,A_27b,V1f_2E0,V2x_2E0) ) ) ).

tff(thm_2Eiterate_2EITERATE__DELETE,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
     => ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3a_2E0: A_27a] :
          ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
            & p(c_2Ebool_2EIN_2E2(A_27a,V3a_2E0,V2s_2E0)) )
         => ( app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),V0op_2E0,app_2E2(A_27a,A_27b,V1f_2E0,V3a_2E0)),c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,c_2Epred__set_2EDELETE_2E2(A_27a,V2s_2E0,V3a_2E0),V1f_2E0)) = c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V2s_2E0,V1f_2E0) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__DELTA,axiom,
    ! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a))))] :
      ( ! [V4x_2E0: A_27b,V2a_2E0: A_27b] :
          ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),F1_2E0,V4x_2E0),V2a_2E0))
        <=> ( V4x_2E0 = V2a_2E0 ) )
     => ( ! [V2a_2E0: A_27b,V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),V4x_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a))),F0_2E0,V2a_2E0),V1f_2E0),V0op_2E0),V4x_2E0) = c_2Ebool_2ECOND_2E3(A_27a,app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),F1_2E0,V4x_2E0),V2a_2E0),app_2E2(A_27b,A_27a,V1f_2E0,V4x_2E0),c_2Eiterate_2Eneutral_2E1(A_27a,V0op_2E0)) )
       => ! [V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))] :
            ( p(c_2Eiterate_2Emonoidal_2E1(A_27a,V0op_2E0))
           => ! [V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V2a_2E0: A_27b,V3s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,V3s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(A_27b,A_27a))),F0_2E0,V2a_2E0),V1f_2E0),V0op_2E0)) = c_2Ebool_2ECOND_2E3(A_27a,c_2Ebool_2EIN_2E2(A_27b,V2a_2E0,V3s_2E0),app_2E2(A_27b,A_27a,V1f_2E0,V2a_2E0),c_2Eiterate_2Eneutral_2E1(A_27a,V0op_2E0)) ) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__IMAGE,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27c,V0op_2E0))
     => ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2g_2E0: tyop_2Emin_2Efun(A_27b,A_27c),V3s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( ! [V4x_2E0: A_27a,V5y_2E0: A_27a] :
              ( ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V3s_2E0))
                & p(c_2Ebool_2EIN_2E2(A_27a,V5y_2E0,V3s_2E0))
                & ( app_2E2(A_27a,A_27b,V1f_2E0,V4x_2E0) = app_2E2(A_27a,A_27b,V1f_2E0,V5y_2E0) ) )
             => ( V4x_2E0 = V5y_2E0 ) )
         => ( c_2Eiterate_2Eiterate_2E3(A_27b,A_27c,V0op_2E0,c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V1f_2E0,V3s_2E0),V2g_2E0) = c_2Eiterate_2Eiterate_2E3(A_27a,A_27c,V0op_2E0,V3s_2E0,c_2Ecombin_2Eo_2E2(A_27a,A_27c,A_27b,V2g_2E0,V1f_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__BIJECTION,axiom,
    ! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V3s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2p_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V5y_2E0: A_27a,V6x_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V3s_2E0),V2p_2E0),V5y_2E0),V6x_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V6x_2E0,V3s_2E0))
            & ( app_2E2(A_27a,A_27a,V2p_2E0,V6x_2E0) = V5y_2E0 ) ) )
     => ! [V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
          ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
         => ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2p_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V3s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
              ( ( ! [V4x_2E0: A_27a] :
                    ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V3s_2E0))
                   => p(c_2Ebool_2EIN_2E2(A_27a,app_2E2(A_27a,A_27a,V2p_2E0,V4x_2E0),V3s_2E0)) )
                & ! [V5y_2E0: A_27a] :
                    ( p(c_2Ebool_2EIN_2E2(A_27a,V5y_2E0,V3s_2E0))
                   => p(c_2Ebool_2E_3F_21_2E1(A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V3s_2E0),V2p_2E0),V5y_2E0))) ) )
             => ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V3s_2E0,V1f_2E0) = c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V3s_2E0,c_2Ecombin_2Eo_2E2(A_27a,A_27b,A_27a,V1f_2E0,V2p_2E0)) ) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__ITERATE__PRODUCT,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,A_27c)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c))),F3_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))))] :
      ( ! [V0op_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c)),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V3x_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V5i_2E0: A_27a] : ( app_2E2(A_27a,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,A_27c)),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,A_27c))),F0_2E0,V0op_2E0),V2t_2E0),V3x_2E0),V5i_2E0) = c_2Eiterate_2Eiterate_2E3(A_27b,A_27c,V0op_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2t_2E0,V5i_2E0),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V3x_2E0,V5i_2E0)) )
     => ( ! [V3x_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V8i_2E0: A_27a,V9j_2E0: A_27b] : ( app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),F2_2E0,V3x_2E0),V8i_2E0),V9j_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V3x_2E0,V8i_2E0),V9j_2E0) )
       => ( ! [V6i_2E0: A_27a,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V7j_2E0: A_27b,V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
              ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool))),F3_2E0,V6i_2E0),V1s_2E0),V7j_2E0),V2t_2E0))
            <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V6i_2E0,V1s_2E0))
                & p(c_2Ebool_2EIN_2E2(A_27b,V7j_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2t_2E0,V6i_2E0))) ) )
         => ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V6i_2E0: A_27a,V7j_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V2t_2E0),V6i_2E0),V7j_2E0) = c_2Epair_2E_2C_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,c_2Epair_2E_2C_2E2(A_27a,A_27b,V6i_2E0,V7j_2E0),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool))),F3_2E0,V6i_2E0),V1s_2E0),V7j_2E0),V2t_2E0)) )
           => ! [V0op_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c))] :
                ( p(c_2Eiterate_2Emonoidal_2E1(A_27c,V0op_2E0))
               => ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V3x_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c))] :
                    ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
                      & ! [V4i_2E0: A_27a] :
                          ( p(c_2Ebool_2EIN_2E2(A_27a,V4i_2E0,V1s_2E0))
                         => p(c_2Epred__set_2EFINITE_2E1(A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2t_2E0,V4i_2E0))) ) )
                   => ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27c,V0op_2E0,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,A_27c)),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,A_27c))),F0_2E0,V0op_2E0),V2t_2E0),V3x_2E0)) = c_2Eiterate_2Eiterate_2E3(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c,V0op_2E0,c_2Epred__set_2EGSPEC_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))),F1_2E0,V1s_2E0),V2t_2E0))),c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),F2_2E0,V3x_2E0))) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__EQ,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
     => ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V3s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( ! [V4x_2E0: A_27a] :
              ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V3s_2E0))
             => ( app_2E2(A_27a,A_27b,V1f_2E0,V4x_2E0) = app_2E2(A_27a,A_27b,V2g_2E0,V4x_2E0) ) )
         => ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V3s_2E0,V1f_2E0) = c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V3s_2E0,V2g_2E0) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__EQ__GENERAL,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V5h_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V6y_2E0: A_27b,V7x_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V1s_2E0),V5h_2E0),V6y_2E0),V7x_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V7x_2E0,V1s_2E0))
            & ( app_2E2(A_27a,A_27b,V5h_2E0,V7x_2E0) = V6y_2E0 ) ) )
     => ! [V0op_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c))] :
          ( p(c_2Eiterate_2Emonoidal_2E1(A_27c,V0op_2E0))
         => ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3f_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V4g_2E0: tyop_2Emin_2Efun(A_27b,A_27c),V5h_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
              ( ( ! [V6y_2E0: A_27b] :
                    ( p(c_2Ebool_2EIN_2E2(A_27b,V6y_2E0,V2t_2E0))
                   => p(c_2Ebool_2E_3F_21_2E1(A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V1s_2E0),V5h_2E0),V6y_2E0))) )
                & ! [V8x_2E0: A_27a] :
                    ( p(c_2Ebool_2EIN_2E2(A_27a,V8x_2E0,V1s_2E0))
                   => ( p(c_2Ebool_2EIN_2E2(A_27b,app_2E2(A_27a,A_27b,V5h_2E0,V8x_2E0),V2t_2E0))
                      & ( app_2E2(A_27b,A_27c,V4g_2E0,app_2E2(A_27a,A_27b,V5h_2E0,V8x_2E0)) = app_2E2(A_27a,A_27c,V3f_2E0,V8x_2E0) ) ) ) )
             => ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27c,V0op_2E0,V1s_2E0,V3f_2E0) = c_2Eiterate_2Eiterate_2E3(A_27b,A_27c,V0op_2E0,V2t_2E0,V4g_2E0) ) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__EQ__GENERAL__INVERSES,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27c,V0op_2E0))
     => ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3f_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V4g_2E0: tyop_2Emin_2Efun(A_27b,A_27c),V5h_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V6k_2E0: tyop_2Emin_2Efun(A_27b,A_27a)] :
          ( ( ! [V7y_2E0: A_27b] :
                ( p(c_2Ebool_2EIN_2E2(A_27b,V7y_2E0,V2t_2E0))
               => ( p(c_2Ebool_2EIN_2E2(A_27a,app_2E2(A_27b,A_27a,V6k_2E0,V7y_2E0),V1s_2E0))
                  & ( app_2E2(A_27a,A_27b,V5h_2E0,app_2E2(A_27b,A_27a,V6k_2E0,V7y_2E0)) = V7y_2E0 ) ) )
            & ! [V8x_2E0: A_27a] :
                ( p(c_2Ebool_2EIN_2E2(A_27a,V8x_2E0,V1s_2E0))
               => ( p(c_2Ebool_2EIN_2E2(A_27b,app_2E2(A_27a,A_27b,V5h_2E0,V8x_2E0),V2t_2E0))
                  & ( app_2E2(A_27b,A_27a,V6k_2E0,app_2E2(A_27a,A_27b,V5h_2E0,V8x_2E0)) = V8x_2E0 )
                  & ( app_2E2(A_27b,A_27c,V4g_2E0,app_2E2(A_27a,A_27b,V5h_2E0,V8x_2E0)) = app_2E2(A_27a,A_27c,V3f_2E0,V8x_2E0) ) ) ) )
         => ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27c,V0op_2E0,V1s_2E0,V3f_2E0) = c_2Eiterate_2Eiterate_2E3(A_27b,A_27c,V0op_2E0,V2t_2E0,V4g_2E0) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__INJECTION,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
     => ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2p_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V3s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V3s_2E0))
            & ! [V4x_2E0: A_27a] :
                ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V3s_2E0))
               => p(c_2Ebool_2EIN_2E2(A_27a,app_2E2(A_27a,A_27a,V2p_2E0,V4x_2E0),V3s_2E0)) )
            & ! [V5x_2E0: A_27a,V6y_2E0: A_27a] :
                ( ( p(c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V3s_2E0))
                  & p(c_2Ebool_2EIN_2E2(A_27a,V6y_2E0,V3s_2E0))
                  & ( app_2E2(A_27a,A_27a,V2p_2E0,V5x_2E0) = app_2E2(A_27a,A_27a,V2p_2E0,V6y_2E0) ) )
               => ( V5x_2E0 = V6y_2E0 ) ) )
         => ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V3s_2E0,c_2Ecombin_2Eo_2E2(A_27a,A_27b,A_27a,V1f_2E0,V2p_2E0)) = c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V3s_2E0,V1f_2E0) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__UNION__NONZERO,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
     => ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
            & p(c_2Epred__set_2EFINITE_2E1(A_27a,V3t_2E0))
            & ! [V4x_2E0: A_27a] :
                ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,c_2Epred__set_2EINTER_2E2(A_27a,V2s_2E0,V3t_2E0)))
               => ( app_2E2(A_27a,A_27b,V1f_2E0,V4x_2E0) = c_2Eiterate_2Eneutral_2E1(A_27b,V0op_2E0) ) ) )
         => ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,c_2Epred__set_2EUNION_2E2(A_27a,V2s_2E0,V3t_2E0),V1f_2E0) = app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),V0op_2E0,c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V2s_2E0,V1f_2E0)),c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V3t_2E0,V1f_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__OP,axiom,
    ! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(A_27b,A_27a))))] :
      ( ! [V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V2g_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V4x_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(A_27b,A_27a),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(A_27b,A_27a)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(A_27b,A_27a))),F0_2E0,V0op_2E0),V1f_2E0),V2g_2E0),V4x_2E0) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,app_2E2(A_27b,A_27a,V1f_2E0,V4x_2E0)),app_2E2(A_27b,A_27a,V2g_2E0,V4x_2E0)) )
     => ! [V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))] :
          ( p(c_2Eiterate_2Emonoidal_2E1(A_27a,V0op_2E0))
         => ! [V1f_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V2g_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V3s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
              ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V3s_2E0))
             => ( c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,V3s_2E0,app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(A_27b,A_27a),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(A_27b,A_27a)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(A_27b,A_27a))),F0_2E0,V0op_2E0),V1f_2E0),V2g_2E0)) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,V3s_2E0,V1f_2E0)),c_2Eiterate_2Eiterate_2E3(A_27b,A_27a,V0op_2E0,V3s_2E0,V2g_2E0)) ) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__SUPERSET,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
     => ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3v_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( ( p(c_2Epred__set_2ESUBSET_2E2(A_27a,V2u_2E0,V3v_2E0))
            & ! [V4x_2E0: A_27a] :
                ( ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V3v_2E0))
                  & ~ p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V2u_2E0)) )
               => ( app_2E2(A_27a,A_27b,V1f_2E0,V4x_2E0) = c_2Eiterate_2Eneutral_2E1(A_27b,V0op_2E0) ) ) )
         => ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V3v_2E0,V1f_2E0) = c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V2u_2E0,V1f_2E0) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__IMAGE__NONZERO,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0op_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27c,A_27c))] :
      ( p(c_2Eiterate_2Emonoidal_2E1(A_27c,V0op_2E0))
     => ! [V1g_2E0: tyop_2Emin_2Efun(A_27b,A_27c),V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V3s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V3s_2E0))
            & ! [V4x_2E0: A_27a,V5y_2E0: A_27a] :
                ( ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V3s_2E0))
                  & p(c_2Ebool_2EIN_2E2(A_27a,V5y_2E0,V3s_2E0))
                  & ( V4x_2E0 != V5y_2E0 )
                  & ( app_2E2(A_27a,A_27b,V2f_2E0,V4x_2E0) = app_2E2(A_27a,A_27b,V2f_2E0,V5y_2E0) ) )
               => ( app_2E2(A_27b,A_27c,V1g_2E0,app_2E2(A_27a,A_27b,V2f_2E0,V4x_2E0)) = c_2Eiterate_2Eneutral_2E1(A_27c,V0op_2E0) ) ) )
         => ( c_2Eiterate_2Eiterate_2E3(A_27b,A_27c,V0op_2E0,c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V2f_2E0,V3s_2E0),V1g_2E0) = c_2Eiterate_2Eiterate_2E3(A_27a,A_27c,V0op_2E0,V3s_2E0,c_2Ecombin_2Eo_2E2(A_27a,A_27c,A_27b,V1g_2E0,V2f_2E0)) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__CASES,axiom,
    ! [A_27a: $tType,A_27b: $tType,F4_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F3_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b))))] :
      ( ! [V7x_2E0: A_27a,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F4_2E0,V7x_2E0),V1s_2E0),V2P_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V7x_2E0,V1s_2E0))
            & ~ p(app_2E2(A_27a,tyop_2Emin_2Ebool,V2P_2E0,V7x_2E0)) ) )
     => ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V7x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F2_2E0,V1s_2E0),V2P_2E0),V7x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V7x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F4_2E0,V7x_2E0),V1s_2E0),V2P_2E0)) )
       => ( ! [V6x_2E0: A_27a,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
              ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F3_2E0,V6x_2E0),V1s_2E0),V2P_2E0))
            <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V6x_2E0,V1s_2E0))
                & p(app_2E2(A_27a,tyop_2Emin_2Ebool,V2P_2E0,V6x_2E0)) ) )
         => ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V6x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V1s_2E0),V2P_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F3_2E0,V6x_2E0),V1s_2E0),V2P_2E0)) )
           => ( ! [V2P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V4g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V5x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b))),F0_2E0,V2P_2E0),V3f_2E0),V4g_2E0),V5x_2E0) = c_2Ebool_2ECOND_2E3(A_27b,app_2E2(A_27a,tyop_2Emin_2Ebool,V2P_2E0,V5x_2E0),app_2E2(A_27a,A_27b,V3f_2E0,V5x_2E0),app_2E2(A_27a,A_27b,V4g_2E0,V5x_2E0)) )
             => ! [V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
                  ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
                 => ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V4g_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
                      ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
                     => ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b))),F0_2E0,V2P_2E0),V3f_2E0),V4g_2E0)) = app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),V0op_2E0,c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V1s_2E0),V2P_2E0)),V3f_2E0)),c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F2_2E0,V1s_2E0),V2P_2E0)),V4g_2E0)) ) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__OP__GEN,axiom,
    ! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b))))] :
      ( ! [V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V4x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b))),F0_2E0,V0op_2E0),V1f_2E0),V2g_2E0),V4x_2E0) = app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),V0op_2E0,app_2E2(A_27a,A_27b,V1f_2E0,V4x_2E0)),app_2E2(A_27a,A_27b,V2g_2E0,V4x_2E0)) )
     => ! [V0op_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))] :
          ( p(c_2Eiterate_2Emonoidal_2E1(A_27b,V0op_2E0))
         => ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V3s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
              ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,c_2Eiterate_2Esupport_2E3(A_27a,A_27b,V0op_2E0,V1f_2E0,V3s_2E0)))
                & p(c_2Epred__set_2EFINITE_2E1(A_27a,c_2Eiterate_2Esupport_2E3(A_27a,A_27b,V0op_2E0,V2g_2E0,V3s_2E0))) )
             => ( c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V3s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b))),F0_2E0,V0op_2E0),V1f_2E0),V2g_2E0)) = app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),V0op_2E0,c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V3s_2E0,V1f_2E0)),c_2Eiterate_2Eiterate_2E3(A_27a,A_27b,V0op_2E0,V3s_2E0,V2g_2E0)) ) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__CLAUSES__NUMSEG,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( ! [V2m_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2m_2E0))
        <=> ( V2m_2E0 = c_2Enum_2E0_2E0 ) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V1op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))] :
          ( p(c_2Eiterate_2Emonoidal_2E1(A_27a,V1op_2E0))
         => ( ! [V2m_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2Eiterate_2E3(tyop_2Enum_2Enum,A_27a,V1op_2E0,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,c_2Enum_2E0_2E0),V0f_2E0) = c_2Ebool_2ECOND_2E3(A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V2m_2E0),app_2E2(tyop_2Enum_2Enum,A_27a,V0f_2E0,c_2Enum_2E0_2E0),c_2Eiterate_2Eneutral_2E1(A_27a,V1op_2E0)) )
            & ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2Eiterate_2E3(tyop_2Enum_2Enum,A_27a,V1op_2E0,c_2Eiterate_2E_2E_2E_2E2(V3m_2E0,c_2Enum_2ESUC_2E1(V4n_2E0)),V0f_2E0) = c_2Ebool_2ECOND_2E3(A_27a,c_2Earithmetic_2E_3C_3D_2E2(V3m_2E0,c_2Enum_2ESUC_2E1(V4n_2E0)),app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V1op_2E0,c_2Eiterate_2Eiterate_2E3(tyop_2Enum_2Enum,A_27a,V1op_2E0,c_2Eiterate_2E_2E_2E_2E2(V3m_2E0,V4n_2E0),V0f_2E0)),app_2E2(tyop_2Enum_2Enum,A_27a,V0f_2E0,c_2Enum_2ESUC_2E1(V4n_2E0))),c_2Eiterate_2Eiterate_2E3(tyop_2Enum_2Enum,A_27a,V1op_2E0,c_2Eiterate_2E_2E_2E_2E2(V3m_2E0,V4n_2E0),V0f_2E0)) ) ) ) ) ).

tff(thm_2Eiterate_2EITERATE__PAIR,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)))] :
      ( ! [V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V4i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,A_27a,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),F0_2E0,V0op_2E0),V1f_2E0),V4i_2E0) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),V0op_2E0,app_2E2(tyop_2Enum_2Enum,A_27a,V1f_2E0,c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V4i_2E0))),app_2E2(tyop_2Enum_2Enum,A_27a,V1f_2E0,c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V4i_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
     => ! [V0op_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))] :
          ( p(c_2Eiterate_2Emonoidal_2E1(A_27a,V0op_2E0))
         => ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2Eiterate_2E3(tyop_2Enum_2Enum,A_27a,V0op_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V2m_2E0),c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V3n_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V1f_2E0) = c_2Eiterate_2Eiterate_2E3(tyop_2Enum_2Enum,A_27a,V0op_2E0,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),F0_2E0,V0op_2E0),V1f_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ENEUTRAL__ADD,axiom,
    c_2Eiterate_2Eneutral_2E1(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E0) = c_2Enum_2E0_2E0 ).

tff(thm_2Eiterate_2ENEUTRAL__MUL,axiom,
    c_2Eiterate_2Eneutral_2E1(tyop_2Enum_2Enum,c_2Earithmetic_2E_2A_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) ).

tff(thm_2Eiterate_2EMONOIDAL__ADD,axiom,
    p(c_2Eiterate_2Emonoidal_2E1(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E0)) ).

tff(thm_2Eiterate_2EMONOIDAL__MUL,axiom,
    p(c_2Eiterate_2Emonoidal_2E1(tyop_2Enum_2Enum,c_2Earithmetic_2E_2A_2E0)) ).

tff(thm_2Eiterate_2ENSUM__DEGENERATE,axiom,
    ! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V2x_2E0: A_27a,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] :
          ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),F1_2E0,V2x_2E0),V1s_2E0),V0f_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
            & ( app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V2x_2E0) != c_2Enum_2E0_2E0 ) ) )
     => ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V1s_2E0),V0f_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V2x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),F1_2E0,V2x_2E0),V1s_2E0),V0f_2E0)) )
       => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
            ( ~ p(c_2Epred__set_2EFINITE_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V1s_2E0),V0f_2E0))))
           => ( c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,V0f_2E0) = c_2Enum_2E0_2E0 ) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__CLAUSES,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] : ( c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EEMPTY_2E0(A_27a),V0f_2E0) = c_2Enum_2E0_2E0 )
      & ! [V1x_2E0: A_27b,V2f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),V3s_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2EFINITE_2E1(A_27b,V3s_2E0))
         => ( c_2Eiterate_2Ensum_2E2(A_27b,c_2Epred__set_2EINSERT_2E2(A_27b,V1x_2E0,V3s_2E0),V2f_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Ebool_2EIN_2E2(A_27b,V1x_2E0,V3s_2E0),c_2Eiterate_2Ensum_2E2(A_27b,V3s_2E0,V2f_2E0),c_2Earithmetic_2E_2B_2E2(app_2E2(A_27b,tyop_2Enum_2Enum,V2f_2E0,V1x_2E0),c_2Eiterate_2Ensum_2E2(A_27b,V3s_2E0,V2f_2E0))) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__UNION,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
        & p(c_2Epred__set_2EFINITE_2E1(A_27a,V2t_2E0))
        & p(c_2Epred__set_2EDISJOINT_2E2(A_27a,V1s_2E0,V2t_2E0)) )
     => ( c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EUNION_2E2(A_27a,V1s_2E0,V2t_2E0),V0f_2E0) = c_2Earithmetic_2E_2B_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,V0f_2E0),c_2Eiterate_2Ensum_2E2(A_27a,V2t_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ENSUM__DIFF,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
        & p(c_2Epred__set_2ESUBSET_2E2(A_27a,V2t_2E0,V1s_2E0)) )
     => ( c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EDIFF_2E2(A_27a,V1s_2E0,V2t_2E0),V0f_2E0) = c_2Earithmetic_2E_2D_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,V0f_2E0),c_2Eiterate_2Ensum_2E2(A_27a,V2t_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ENSUM__INCL__EXCL,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
        & p(c_2Epred__set_2EFINITE_2E1(A_27a,V1t_2E0)) )
     => ( c_2Earithmetic_2E_2B_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V0s_2E0,V2f_2E0),c_2Eiterate_2Ensum_2E2(A_27a,V1t_2E0,V2f_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EUNION_2E2(A_27a,V0s_2E0,V1t_2E0),V2f_2E0),c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EINTER_2E2(A_27a,V0s_2E0,V1t_2E0),V2f_2E0)) ) ) ).

tff(thm_2Eiterate_2ENSUM__SUPPORT,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Ensum_2E2(A_27a,c_2Eiterate_2Esupport_2E3(A_27a,tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E0,V0f_2E0,V1s_2E0),V0f_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,V0f_2E0) ) ).

tff(thm_2Eiterate_2ENSUM__ADD,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V0f_2E0),V1g_2E0),V3x_2E0) = c_2Earithmetic_2E_2B_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Enum_2Enum,V1g_2E0,V3x_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
         => ( c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V0f_2E0),V1g_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V0f_2E0),c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V1g_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__ADD__GEN,axiom,
    ! [A_27a: $tType,F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum))),F4_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F3_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F2_2E0,V0f_2E0),V1g_2E0),V5x_2E0) = c_2Earithmetic_2E_2B_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V5x_2E0),app_2E2(A_27a,tyop_2Enum_2Enum,V1g_2E0,V5x_2E0)) )
     => ( ! [V4x_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] :
            ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),F4_2E0,V4x_2E0),V2s_2E0),V1g_2E0))
          <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V2s_2E0))
              & ( app_2E2(A_27a,tyop_2Enum_2Enum,V1g_2E0,V4x_2E0) != c_2Enum_2E0_2E0 ) ) )
       => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V2s_2E0),V1g_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),F4_2E0,V4x_2E0),V2s_2E0),V1g_2E0)) )
         => ( ! [V3x_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] :
                ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),F3_2E0,V3x_2E0),V2s_2E0),V0f_2E0))
              <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
                  & ( app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0) != c_2Enum_2E0_2E0 ) ) )
           => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V2s_2E0),V0f_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),F3_2E0,V3x_2E0),V2s_2E0),V0f_2E0)) )
             => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
                  ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V2s_2E0),V0f_2E0))))
                    & p(c_2Epred__set_2EFINITE_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V2s_2E0),V1g_2E0)))) )
                 => ( c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F2_2E0,V0f_2E0),V1g_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V0f_2E0),c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V1g_2E0)) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__EQ__0,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ! [V2x_2E0: A_27a] :
          ( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
         => ( app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V2x_2E0) = c_2Enum_2E0_2E0 ) )
     => ( c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,V0f_2E0) = c_2Enum_2E0_2E0 ) ) ).

tff(thm_2Eiterate_2ENSUM__0,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] :
      ( ! [V1n_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,F0_2E0,V1n_2E0) = c_2Enum_2E0_2E0 )
     => ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Ensum_2E2(A_27a,V0s_2E0,F0_2E0) = c_2Enum_2E0_2E0 ) ) ).

tff(thm_2Eiterate_2ENSUM__LMUL,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)))] :
      ( ! [V1c_2E0: tyop_2Enum_2Enum,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V1c_2E0),V0f_2E0),V3x_2E0) = c_2Earithmetic_2E_2A_2E2(V1c_2E0,app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1c_2E0: tyop_2Enum_2Enum,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V1c_2E0),V0f_2E0)) = c_2Earithmetic_2E_2A_2E2(V1c_2E0,c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ENSUM__RMUL,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1c_2E0: tyop_2Enum_2Enum,V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V0f_2E0),V1c_2E0),V3x_2E0) = c_2Earithmetic_2E_2A_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0),V1c_2E0) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1c_2E0: tyop_2Enum_2Enum,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V0f_2E0),V1c_2E0)) = c_2Earithmetic_2E_2A_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V0f_2E0),V1c_2E0) ) ) ).

tff(thm_2Eiterate_2ENSUM__LE,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
           => p(c_2Earithmetic_2E_3C_3D_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Enum_2Enum,V1g_2E0,V3x_2E0))) ) )
     => p(c_2Earithmetic_2E_3C_3D_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V0f_2E0),c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V1g_2E0))) ) ).

tff(thm_2Eiterate_2ENSUM__LT,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
           => p(c_2Earithmetic_2E_3C_3D_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Enum_2Enum,V1g_2E0,V3x_2E0))) )
        & ? [V4x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V2s_2E0))
            & p(c_2Eprim__rec_2E_3C_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V4x_2E0),app_2E2(A_27a,tyop_2Enum_2Enum,V1g_2E0,V4x_2E0))) ) )
     => p(c_2Eprim__rec_2E_3C_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V0f_2E0),c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V1g_2E0))) ) ).

tff(thm_2Eiterate_2ENSUM__LT__ALL,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
        & ( V2s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27a) )
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
           => p(c_2Eprim__rec_2E_3C_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Enum_2Enum,V1g_2E0,V3x_2E0))) ) )
     => p(c_2Eprim__rec_2E_3C_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V0f_2E0),c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V1g_2E0))) ) ).

tff(thm_2Eiterate_2ENSUM__EQ,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ! [V3x_2E0: A_27a] :
          ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
         => ( app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0) = app_2E2(A_27a,tyop_2Enum_2Enum,V1g_2E0,V3x_2E0) ) )
     => ( c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V0f_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V1g_2E0) ) ) ).

tff(thm_2Eiterate_2ENSUM__CONST,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum))] :
      ( ! [V0c_2E0: tyop_2Enum_2Enum,V2n_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),F0_2E0,V0c_2E0),V2n_2E0) = V0c_2E0 )
     => ! [V0c_2E0: tyop_2Enum_2Enum,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
         => ( c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),F0_2E0,V0c_2E0)) = c_2Earithmetic_2E_2A_2E2(c_2Epred__set_2ECARD_2E1(A_27a,V1s_2E0),V0c_2E0) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__POS__BOUND,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1b_2E0: tyop_2Enum_2Enum,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
        & p(c_2Earithmetic_2E_3C_3D_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V0f_2E0),V1b_2E0)) )
     => ! [V3x_2E0: A_27a] :
          ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
         => p(c_2Earithmetic_2E_3C_3D_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0),V1b_2E0)) ) ) ).

tff(thm_2Eiterate_2ENSUM__EQ__0__IFF,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
     => ( ( c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,V0f_2E0) = c_2Enum_2E0_2E0 )
      <=> ! [V2x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
           => ( app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V2x_2E0) = c_2Enum_2E0_2E0 ) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__POS__LT,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
        & ? [V2x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0))
            & p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V2x_2E0))) ) )
     => p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,V0f_2E0))) ) ).

tff(thm_2Eiterate_2ENSUM__POS__LT__ALL,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27a) )
        & ! [V2i_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V2i_2E0,V0s_2E0))
           => p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,app_2E2(A_27a,tyop_2Enum_2Enum,V1f_2E0,V2i_2E0))) ) )
     => p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,c_2Eiterate_2Ensum_2E2(A_27a,V0s_2E0,V1f_2E0))) ) ).

tff(thm_2Eiterate_2ENSUM__DELETE,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2a_2E0: A_27a] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
        & p(c_2Ebool_2EIN_2E2(A_27a,V2a_2E0,V1s_2E0)) )
     => ( c_2Earithmetic_2E_2B_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V2a_2E0),c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EDELETE_2E2(A_27a,V1s_2E0,V2a_2E0),V0f_2E0)) = c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,V0f_2E0) ) ) ).

tff(thm_2Eiterate_2ENSUM__SING,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1x_2E0: A_27a] : ( c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EINSERT_2E2(A_27a,V1x_2E0,c_2Epred__set_2EEMPTY_2E0(A_27a)),V0f_2E0) = app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V1x_2E0) ) ).

tff(thm_2Eiterate_2ENSUM__DELTA,axiom,
    ! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)))] :
      ( ! [V3x_2E0: A_27a,V2a_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F1_2E0,V3x_2E0),V2a_2E0))
        <=> ( V3x_2E0 = V2a_2E0 ) )
     => ( ! [V2a_2E0: A_27a,V0b_2E0: tyop_2Enum_2Enum,V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V2a_2E0),V0b_2E0),V3x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),F1_2E0,V3x_2E0),V2a_2E0),V0b_2E0,c_2Enum_2E0_2E0) )
       => ! [V0b_2E0: tyop_2Enum_2Enum,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2a_2E0: A_27a] : ( c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V2a_2E0),V0b_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Ebool_2EIN_2E2(A_27a,V2a_2E0,V1s_2E0),V0b_2E0,c_2Enum_2E0_2E0) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__SWAP,axiom,
    ! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)))] :
      ( ! [V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),V3i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V2t_2E0),V0f_2E0),V3i_2E0) = c_2Eiterate_2Ensum_2E2(A_27b,V2t_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),V0f_2E0,V3i_2E0)) )
     => ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),V4j_2E0: A_27b,V5i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F2_2E0,V0f_2E0),V4j_2E0),V5i_2E0) = app_2E2(A_27b,tyop_2Enum_2Enum,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),V0f_2E0,V5i_2E0),V4j_2E0) )
       => ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),V4j_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),F1_2E0,V1s_2E0),V0f_2E0),V4j_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F2_2E0,V0f_2E0),V4j_2E0)) )
         => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
              ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
                & p(c_2Epred__set_2EFINITE_2E1(A_27b,V2t_2E0)) )
             => ( c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V2t_2E0),V0f_2E0)) = c_2Eiterate_2Ensum_2E2(A_27b,V2t_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),F1_2E0,V1s_2E0),V0f_2E0)) ) ) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__IMAGE,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ! [V3x_2E0: A_27a,V4y_2E0: A_27a] :
          ( ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
            & p(c_2Ebool_2EIN_2E2(A_27a,V4y_2E0,V2s_2E0))
            & ( app_2E2(A_27a,A_27b,V0f_2E0,V3x_2E0) = app_2E2(A_27a,A_27b,V0f_2E0,V4y_2E0) ) )
         => ( V3x_2E0 = V4y_2E0 ) )
     => ( c_2Eiterate_2Ensum_2E2(A_27b,c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V0f_2E0,V2s_2E0),V1g_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Enum_2Enum,A_27b,V1g_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ENSUM__SUPERSET,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2v_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2ESUBSET_2E2(A_27a,V1u_2E0,V2v_2E0))
        & ! [V3x_2E0: A_27a] :
            ( ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2v_2E0))
              & ~ p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1u_2E0)) )
           => ( app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0) = c_2Enum_2E0_2E0 ) ) )
     => ( c_2Eiterate_2Ensum_2E2(A_27a,V2v_2E0,V0f_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,V1u_2E0,V0f_2E0) ) ) ).

tff(thm_2Eiterate_2ENSUM__UNION__RZERO,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2v_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1u_2E0))
        & ! [V3x_2E0: A_27a] :
            ( ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2v_2E0))
              & ~ p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1u_2E0)) )
           => ( app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0) = c_2Enum_2E0_2E0 ) ) )
     => ( c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EUNION_2E2(A_27a,V1u_2E0,V2v_2E0),V0f_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,V1u_2E0,V0f_2E0) ) ) ).

tff(thm_2Eiterate_2ENSUM__UNION__LZERO,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2v_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2v_2E0))
        & ! [V3x_2E0: A_27a] :
            ( ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1u_2E0))
              & ~ p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2v_2E0)) )
           => ( app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0) = c_2Enum_2E0_2E0 ) ) )
     => ( c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EUNION_2E2(A_27a,V1u_2E0,V2v_2E0),V0f_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,V2v_2E0,V0f_2E0) ) ) ).

tff(thm_2Eiterate_2ENSUM__RESTRICT,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)))] :
      ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V1s_2E0),V0f_2E0),V2x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Ebool_2EIN_2E2(A_27a,V2x_2E0,V1s_2E0),app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V2x_2E0),c_2Enum_2E0_2E0) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
         => ( c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V1s_2E0),V0f_2E0)) = c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,V0f_2E0) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__BOUND,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2b_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
           => p(c_2Earithmetic_2E_3C_3D_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V1f_2E0,V3x_2E0),V2b_2E0)) ) )
     => p(c_2Earithmetic_2E_3C_3D_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V0s_2E0,V1f_2E0),c_2Earithmetic_2E_2A_2E2(c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0),V2b_2E0))) ) ).

tff(thm_2Eiterate_2ENSUM__BOUND__GEN,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2b_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27a) )
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
           => p(c_2Earithmetic_2E_3C_3D_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V1f_2E0,V3x_2E0),c_2Earithmetic_2EDIV_2E2(V2b_2E0,c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0)))) ) )
     => p(c_2Earithmetic_2E_3C_3D_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V0s_2E0,V1f_2E0),V2b_2E0)) ) ).

tff(thm_2Eiterate_2ENSUM__BOUND__LT,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2b_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
           => p(c_2Earithmetic_2E_3C_3D_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V1f_2E0,V3x_2E0),V2b_2E0)) )
        & ? [V4x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V0s_2E0))
            & p(c_2Eprim__rec_2E_3C_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V1f_2E0,V4x_2E0),V2b_2E0)) ) )
     => p(c_2Eprim__rec_2E_3C_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V0s_2E0,V1f_2E0),c_2Earithmetic_2E_2A_2E2(c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0),V2b_2E0))) ) ).

tff(thm_2Eiterate_2ENSUM__BOUND__LT__ALL,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2b_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27a) )
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
           => p(c_2Eprim__rec_2E_3C_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V1f_2E0,V3x_2E0),V2b_2E0)) ) )
     => p(c_2Eprim__rec_2E_3C_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V0s_2E0,V1f_2E0),c_2Earithmetic_2E_2A_2E2(c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0),V2b_2E0))) ) ).

tff(thm_2Eiterate_2ENSUM__BOUND__LT__GEN,axiom,
    ! [A_27a: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2b_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
        & ( V0s_2E0 != c_2Epred__set_2EEMPTY_2E0(A_27a) )
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V0s_2E0))
           => p(c_2Eprim__rec_2E_3C_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V1f_2E0,V3x_2E0),c_2Earithmetic_2EDIV_2E2(V2b_2E0,c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0)))) ) )
     => p(c_2Eprim__rec_2E_3C_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V0s_2E0,V1f_2E0),V2b_2E0)) ) ).

tff(thm_2Eiterate_2ENSUM__UNION__EQ,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V3u_2E0))
        & ( c_2Epred__set_2EINTER_2E2(A_27a,V1s_2E0,V2t_2E0) = c_2Epred__set_2EEMPTY_2E0(A_27a) )
        & ( c_2Epred__set_2EUNION_2E2(A_27a,V1s_2E0,V2t_2E0) = V3u_2E0 ) )
     => ( c_2Earithmetic_2E_2B_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,V0f_2E0),c_2Eiterate_2Ensum_2E2(A_27a,V2t_2E0,V0f_2E0)) = c_2Eiterate_2Ensum_2E2(A_27a,V3u_2E0,V0f_2E0) ) ) ).

tff(thm_2Eiterate_2ENSUM__EQ__SUPERSET,axiom,
    ! [A_27a: $tType,V0g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V3t_2E0))
        & p(c_2Epred__set_2ESUBSET_2E2(A_27a,V3t_2E0,V2s_2E0))
        & ! [V4x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V3t_2E0))
           => ( app_2E2(A_27a,tyop_2Enum_2Enum,V1f_2E0,V4x_2E0) = app_2E2(A_27a,tyop_2Enum_2Enum,V0g_2E0,V4x_2E0) ) )
        & ! [V5x_2E0: A_27a] :
            ( ( p(c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V2s_2E0))
              & ~ p(c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V3t_2E0)) )
           => ( app_2E2(A_27a,tyop_2Enum_2Enum,V1f_2E0,V5x_2E0) = c_2Enum_2E0_2E0 ) ) )
     => ( c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V1f_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,V3t_2E0,V0g_2E0) ) ) ).

tff(thm_2Eiterate_2ENSUM__RESTRICT__SET,axiom,
    ! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum))),F2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F1_2E0,V0P_2E0),V2f_2E0),V4x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V4x_2E0),app_2E2(A_27a,tyop_2Enum_2Enum,V2f_2E0,V4x_2E0),c_2Enum_2E0_2E0) )
     => ( ! [V3x_2E0: A_27a,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
            ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F2_2E0,V3x_2E0),V1s_2E0),V0P_2E0))
          <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V1s_2E0))
              & p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0)) ) )
       => ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V1s_2E0),V0P_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F2_2E0,V3x_2E0),V1s_2E0),V0P_2E0)) )
         => ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] : ( c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V1s_2E0),V0P_2E0)),V2f_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F1_2E0,V0P_2E0),V2f_2E0)) ) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__NSUM__RESTRICT,axiom,
    ! [A_27a: $tType,A_27b: $tType,F7_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F5_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F6_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)))),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum))),F4_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum))))] :
      ( ! [V5y_2E0: A_27b,V3t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V4x_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F7_2E0,V5y_2E0),V3t_2E0),V0R_2E0),V4x_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27b,V5y_2E0,V3t_2E0))
            & p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0,V4x_2E0),V5y_2E0)) ) )
     => ( ! [V3t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V4x_2E0: A_27a,V5y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F5_2E0,V3t_2E0),V0R_2E0),V4x_2E0),V5y_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,V5y_2E0,app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F7_2E0,V5y_2E0),V3t_2E0),V0R_2E0),V4x_2E0)) )
       => ( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),V4x_2E0: A_27a,V6y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Enum_2Enum,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),F6_2E0,V1f_2E0),V4x_2E0),V6y_2E0) = app_2E2(A_27b,tyop_2Enum_2Enum,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),V1f_2E0,V4x_2E0),V6y_2E0) )
         => ( ! [V3t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum))),F0_2E0,V3t_2E0),V0R_2E0),V1f_2E0),V4x_2E0) = c_2Eiterate_2Ensum_2E2(A_27b,c_2Epred__set_2EGSPEC_2E1(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F5_2E0,V3t_2E0),V0R_2E0),V4x_2E0)),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),F6_2E0,V1f_2E0),V4x_2E0)) )
           => ( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),V7y_2E0: A_27b,V9x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F3_2E0,V1f_2E0),V7y_2E0),V9x_2E0) = app_2E2(A_27b,tyop_2Enum_2Enum,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),V1f_2E0,V9x_2E0),V7y_2E0) )
             => ( ! [V8x_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V7y_2E0: A_27b] :
                    ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F4_2E0,V8x_2E0),V2s_2E0),V0R_2E0),V7y_2E0))
                  <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V8x_2E0,V2s_2E0))
                      & p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0,V8x_2E0),V7y_2E0)) ) )
               => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V7y_2E0: A_27b,V8x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F2_2E0,V2s_2E0),V0R_2E0),V7y_2E0),V8x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V8x_2E0,app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F4_2E0,V8x_2E0),V2s_2E0),V0R_2E0),V7y_2E0)) )
                 => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),V7y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum))),F1_2E0,V2s_2E0),V0R_2E0),V1f_2E0),V7y_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F2_2E0,V2s_2E0),V0R_2E0),V7y_2E0)),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F3_2E0,V1f_2E0),V7y_2E0)) )
                   => ! [V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
                        ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
                          & p(c_2Epred__set_2EFINITE_2E1(A_27b,V3t_2E0)) )
                       => ( c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum))),F0_2E0,V3t_2E0),V0R_2E0),V1f_2E0)) = c_2Eiterate_2Ensum_2E2(A_27b,V3t_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum))),F1_2E0,V2s_2E0),V0R_2E0),V1f_2E0)) ) ) ) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2ECARD__EQ__NSUM,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] :
      ( ! [V1x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,F0_2E0,V1x_2E0) = c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)) )
     => ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
         => ( c_2Epred__set_2ECARD_2E1(A_27a,V0s_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,V0s_2E0,F0_2E0) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__MULTICOUNT__GEN,axiom,
    ! [A_27a: $tType,A_27b: $tType,F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),F5_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F4_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum))),F3_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))))] :
      ( ! [V3k_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),V8i_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),F2_2E0,V3k_2E0),V8i_2E0) = app_2E2(A_27b,tyop_2Enum_2Enum,V3k_2E0,V8i_2E0) )
     => ( ! [V7j_2E0: A_27b,V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V6i_2E0: A_27a] :
            ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F5_2E0,V7j_2E0),V2t_2E0),V0R_2E0),V6i_2E0))
          <=> ( p(c_2Ebool_2EIN_2E2(A_27b,V7j_2E0,V2t_2E0))
              & p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0,V6i_2E0),V7j_2E0)) ) )
       => ( ! [V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V6i_2E0: A_27a,V7j_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F4_2E0,V2t_2E0),V0R_2E0),V6i_2E0),V7j_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,V7j_2E0,app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F5_2E0,V7j_2E0),V2t_2E0),V0R_2E0),V6i_2E0)) )
         => ( ! [V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V6i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F1_2E0,V2t_2E0),V0R_2E0),V6i_2E0) = c_2Epred__set_2ECARD_2E1(A_27b,c_2Epred__set_2EGSPEC_2E1(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F4_2E0,V2t_2E0),V0R_2E0),V6i_2E0))) )
           => ( ! [V5i_2E0: A_27a,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V4j_2E0: A_27b] :
                  ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F3_2E0,V5i_2E0),V1s_2E0),V0R_2E0),V4j_2E0))
                <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V5i_2E0,V1s_2E0))
                    & p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0,V5i_2E0),V4j_2E0)) ) )
             => ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V4j_2E0: A_27b,V5i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0R_2E0),V4j_2E0),V5i_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V5i_2E0,app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F3_2E0,V5i_2E0),V1s_2E0),V0R_2E0),V4j_2E0)) )
               => ! [V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3k_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)] :
                    ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
                      & p(c_2Epred__set_2EFINITE_2E1(A_27b,V2t_2E0))
                      & ! [V4j_2E0: A_27b] :
                          ( p(c_2Ebool_2EIN_2E2(A_27b,V4j_2E0,V2t_2E0))
                         => ( c_2Epred__set_2ECARD_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0R_2E0),V4j_2E0))) = app_2E2(A_27b,tyop_2Enum_2Enum,V3k_2E0,V4j_2E0) ) ) )
                   => ( c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F1_2E0,V2t_2E0),V0R_2E0)) = c_2Eiterate_2Ensum_2E2(A_27b,V2t_2E0,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),F2_2E0,V3k_2E0)) ) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__MULTICOUNT,axiom,
    ! [A_27a: $tType,A_27b: $tType,F4_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum))),F2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))))] :
      ( ! [V7j_2E0: A_27b,V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V6i_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F4_2E0,V7j_2E0),V2t_2E0),V0R_2E0),V6i_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27b,V7j_2E0,V2t_2E0))
            & p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0,V6i_2E0),V7j_2E0)) ) )
     => ( ! [V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V6i_2E0: A_27a,V7j_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F3_2E0,V2t_2E0),V0R_2E0),V6i_2E0),V7j_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,V7j_2E0,app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F4_2E0,V7j_2E0),V2t_2E0),V0R_2E0),V6i_2E0)) )
       => ( ! [V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V6i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F1_2E0,V2t_2E0),V0R_2E0),V6i_2E0) = c_2Epred__set_2ECARD_2E1(A_27b,c_2Epred__set_2EGSPEC_2E1(A_27b,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F3_2E0,V2t_2E0),V0R_2E0),V6i_2E0))) )
         => ( ! [V5i_2E0: A_27a,V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V4j_2E0: A_27b] :
                ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F2_2E0,V5i_2E0),V1s_2E0),V0R_2E0),V4j_2E0))
              <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V5i_2E0,V1s_2E0))
                  & p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0R_2E0,V5i_2E0),V4j_2E0)) ) )
           => ( ! [V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V4j_2E0: A_27b,V5i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0R_2E0),V4j_2E0),V5i_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V5i_2E0,app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F2_2E0,V5i_2E0),V1s_2E0),V0R_2E0),V4j_2E0)) )
             => ! [V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3k_2E0: tyop_2Enum_2Enum] :
                  ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
                    & p(c_2Epred__set_2EFINITE_2E1(A_27b,V2t_2E0))
                    & ! [V4j_2E0: A_27b] :
                        ( p(c_2Ebool_2EIN_2E2(A_27b,V4j_2E0,V2t_2E0))
                       => ( c_2Epred__set_2ECARD_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F0_2E0,V1s_2E0),V0R_2E0),V4j_2E0))) = V3k_2E0 ) ) )
                 => ( c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F1_2E0,V2t_2E0),V0R_2E0)) = c_2Earithmetic_2E_2A_2E2(V3k_2E0,c_2Epred__set_2ECARD_2E1(A_27b,V2t_2E0)) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__IMAGE__GEN,axiom,
    ! [A_27a: $tType,A_27b: $tType,F2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum))))] :
      ( ! [V4x_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V3y_2E0: A_27b] :
          ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V2s_2E0),V0f_2E0),V3y_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V2s_2E0))
            & ( app_2E2(A_27a,A_27b,V0f_2E0,V4x_2E0) = V3y_2E0 ) ) )
     => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V3y_2E0: A_27b,V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V3y_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F2_2E0,V4x_2E0),V2s_2E0),V0f_2E0),V3y_2E0)) )
       => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V3y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum))),F0_2E0,V2s_2E0),V0f_2E0),V1g_2E0),V3y_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V3y_2E0)),V1g_2E0) )
         => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
              ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
             => ( c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V1g_2E0) = c_2Eiterate_2Ensum_2E2(A_27b,c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V0f_2E0,V2s_2E0),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum))),F0_2E0,V2s_2E0),V0f_2E0),V1g_2E0)) ) ) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__GROUP,axiom,
    ! [A_27a: $tType,A_27b: $tType,F2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum))))] :
      ( ! [V5x_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V4y_2E0: A_27b] :
          ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F2_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4y_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V2s_2E0))
            & ( app_2E2(A_27a,A_27b,V0f_2E0,V5x_2E0) = V4y_2E0 ) ) )
     => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V4y_2E0: A_27b,V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V4y_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F2_2E0,V5x_2E0),V2s_2E0),V0f_2E0),V4y_2E0)) )
       => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V4y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum))),F0_2E0,V2s_2E0),V0f_2E0),V1g_2E0),V4y_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F1_2E0,V2s_2E0),V0f_2E0),V4y_2E0)),V1g_2E0) )
         => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
              ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
                & p(c_2Epred__set_2ESUBSET_2E2(A_27b,c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V0f_2E0,V2s_2E0),V3t_2E0)) )
             => ( c_2Eiterate_2Ensum_2E2(A_27b,V3t_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum))),F0_2E0,V2s_2E0),V0f_2E0),V1g_2E0)) = c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V1g_2E0) ) ) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__SUBSET,axiom,
    ! [A_27a: $tType,V0u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1v_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0u_2E0))
        & p(c_2Epred__set_2EFINITE_2E1(A_27a,V1v_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,c_2Epred__set_2EDIFF_2E2(A_27a,V0u_2E0,V1v_2E0)))
           => ( app_2E2(A_27a,tyop_2Enum_2Enum,V2f_2E0,V3x_2E0) = c_2Enum_2E0_2E0 ) ) )
     => p(c_2Earithmetic_2E_3C_3D_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V0u_2E0,V2f_2E0),c_2Eiterate_2Ensum_2E2(A_27a,V1v_2E0,V2f_2E0))) ) ).

tff(thm_2Eiterate_2ENSUM__SUBSET__SIMPLE,axiom,
    ! [A_27a: $tType,V0u_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1v_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1v_2E0))
        & p(c_2Epred__set_2ESUBSET_2E2(A_27a,V0u_2E0,V1v_2E0)) )
     => p(c_2Earithmetic_2E_3C_3D_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V0u_2E0,V2f_2E0),c_2Eiterate_2Ensum_2E2(A_27a,V1v_2E0,V2f_2E0))) ) ).

tff(thm_2Eiterate_2ENSUM__LE__GEN,axiom,
    ! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V4x_2E0: A_27a,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] :
          ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),F1_2E0,V4x_2E0),V2s_2E0),V1g_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V2s_2E0))
            & ( app_2E2(A_27a,tyop_2Enum_2Enum,V1g_2E0,V4x_2E0) != c_2Enum_2E0_2E0 ) ) )
     => ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V2s_2E0),V1g_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V4x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Ebool)),F1_2E0,V4x_2E0),V2s_2E0),V1g_2E0)) )
       => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
            ( ( ! [V3x_2E0: A_27a] :
                  ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
                 => p(c_2Earithmetic_2E_3C_3D_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0),app_2E2(A_27a,tyop_2Enum_2Enum,V1g_2E0,V3x_2E0))) )
              & p(c_2Epred__set_2EFINITE_2E1(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V2s_2E0),V1g_2E0)))) )
           => p(c_2Earithmetic_2E_3C_3D_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V0f_2E0),c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V1g_2E0))) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__IMAGE__NONZERO,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0d_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),V1i_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
        & ! [V3x_2E0: A_27a,V4y_2E0: A_27a] :
            ( ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
              & p(c_2Ebool_2EIN_2E2(A_27a,V4y_2E0,V2s_2E0))
              & ( V3x_2E0 != V4y_2E0 )
              & ( app_2E2(A_27a,A_27b,V1i_2E0,V3x_2E0) = app_2E2(A_27a,A_27b,V1i_2E0,V4y_2E0) ) )
           => ( app_2E2(A_27b,tyop_2Enum_2Enum,V0d_2E0,app_2E2(A_27a,A_27b,V1i_2E0,V3x_2E0)) = c_2Enum_2E0_2E0 ) ) )
     => ( c_2Eiterate_2Ensum_2E2(A_27b,c_2Epred__set_2EIMAGE_2E2(A_27a,A_27b,V1i_2E0,V2s_2E0),V0d_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Enum_2Enum,A_27b,V0d_2E0,V1i_2E0)) ) ) ).

tff(thm_2Eiterate_2ENSUM__BIJECTION,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1p_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V4y_2E0: A_27a,V5x_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V2s_2E0),V1p_2E0),V4y_2E0),V5x_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V2s_2E0))
            & ( app_2E2(A_27a,A_27a,V1p_2E0,V5x_2E0) = V4y_2E0 ) ) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1p_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( ( ! [V3x_2E0: A_27a] :
                ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
               => p(c_2Ebool_2EIN_2E2(A_27a,app_2E2(A_27a,A_27a,V1p_2E0,V3x_2E0),V2s_2E0)) )
            & ! [V4y_2E0: A_27a] :
                ( p(c_2Ebool_2EIN_2E2(A_27a,V4y_2E0,V2s_2E0))
               => p(c_2Ebool_2E_3F_21_2E1(A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V2s_2E0),V1p_2E0),V4y_2E0))) ) )
         => ( c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V0f_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Enum_2Enum,A_27a,V0f_2E0,V1p_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__NSUM__PRODUCT,axiom,
    ! [A_27a: $tType,A_27b: $tType,F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum))),F3_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))))] :
      ( ! [V2x_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),V7i_2E0: A_27a,V8j_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Enum_2Enum,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),F2_2E0,V2x_2E0),V7i_2E0),V8j_2E0) = app_2E2(A_27b,tyop_2Enum_2Enum,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),V2x_2E0,V7i_2E0),V8j_2E0) )
     => ( ! [V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V2x_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),V4i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V1t_2E0),V2x_2E0),V4i_2E0) = c_2Eiterate_2Ensum_2E2(A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1t_2E0,V4i_2E0),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),V2x_2E0,V4i_2E0)) )
       => ( ! [V5i_2E0: A_27a,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V6j_2E0: A_27b,V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
              ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool))),F3_2E0,V5i_2E0),V0s_2E0),V6j_2E0),V1t_2E0))
            <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V5i_2E0,V0s_2E0))
                & p(c_2Ebool_2EIN_2E2(A_27b,V6j_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1t_2E0,V5i_2E0))) ) )
         => ( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V5i_2E0: A_27a,V6j_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))),F1_2E0,V0s_2E0),V1t_2E0),V5i_2E0),V6j_2E0) = c_2Epair_2E_2C_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,c_2Epair_2E_2C_2E2(A_27a,A_27b,V5i_2E0,V6j_2E0),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool))),F3_2E0,V5i_2E0),V0s_2E0),V6j_2E0),V1t_2E0)) )
           => ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V2x_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum))] :
                ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
                  & ! [V3i_2E0: A_27a] :
                      ( p(c_2Ebool_2EIN_2E2(A_27a,V3i_2E0,V0s_2E0))
                     => p(c_2Epred__set_2EFINITE_2E1(A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1t_2E0,V3i_2E0))) ) )
               => ( c_2Eiterate_2Ensum_2E2(A_27a,V0s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V1t_2E0),V2x_2E0)) = c_2Eiterate_2Ensum_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epred__set_2EGSPEC_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))),F1_2E0,V0s_2E0),V1t_2E0))),c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum)),F2_2E0,V2x_2E0))) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__EQ__GENERAL,axiom,
    ! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4h_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V5y_2E0: A_27b,V6x_2E0: A_27a] :
          ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V4h_2E0),V5y_2E0),V6x_2E0))
        <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V6x_2E0,V0s_2E0))
            & ( app_2E2(A_27a,A_27b,V4h_2E0,V6x_2E0) = V5y_2E0 ) ) )
     => ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V3g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),V4h_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
          ( ( ! [V5y_2E0: A_27b] :
                ( p(c_2Ebool_2EIN_2E2(A_27b,V5y_2E0,V1t_2E0))
               => p(c_2Ebool_2E_3F_21_2E1(A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0s_2E0),V4h_2E0),V5y_2E0))) )
            & ! [V7x_2E0: A_27a] :
                ( p(c_2Ebool_2EIN_2E2(A_27a,V7x_2E0,V0s_2E0))
               => ( p(c_2Ebool_2EIN_2E2(A_27b,app_2E2(A_27a,A_27b,V4h_2E0,V7x_2E0),V1t_2E0))
                  & ( app_2E2(A_27b,tyop_2Enum_2Enum,V3g_2E0,app_2E2(A_27a,A_27b,V4h_2E0,V7x_2E0)) = app_2E2(A_27a,tyop_2Enum_2Enum,V2f_2E0,V7x_2E0) ) ) ) )
         => ( c_2Eiterate_2Ensum_2E2(A_27a,V0s_2E0,V2f_2E0) = c_2Eiterate_2Ensum_2E2(A_27b,V1t_2E0,V3g_2E0) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__EQ__GENERAL__INVERSES,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1t_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V3g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Enum_2Enum),V4h_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V5k_2E0: tyop_2Emin_2Efun(A_27b,A_27a)] :
      ( ( ! [V6y_2E0: A_27b] :
            ( p(c_2Ebool_2EIN_2E2(A_27b,V6y_2E0,V1t_2E0))
           => ( p(c_2Ebool_2EIN_2E2(A_27a,app_2E2(A_27b,A_27a,V5k_2E0,V6y_2E0),V0s_2E0))
              & ( app_2E2(A_27a,A_27b,V4h_2E0,app_2E2(A_27b,A_27a,V5k_2E0,V6y_2E0)) = V6y_2E0 ) ) )
        & ! [V7x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V7x_2E0,V0s_2E0))
           => ( p(c_2Ebool_2EIN_2E2(A_27b,app_2E2(A_27a,A_27b,V4h_2E0,V7x_2E0),V1t_2E0))
              & ( app_2E2(A_27b,A_27a,V5k_2E0,app_2E2(A_27a,A_27b,V4h_2E0,V7x_2E0)) = V7x_2E0 )
              & ( app_2E2(A_27b,tyop_2Enum_2Enum,V3g_2E0,app_2E2(A_27a,A_27b,V4h_2E0,V7x_2E0)) = app_2E2(A_27a,tyop_2Enum_2Enum,V2f_2E0,V7x_2E0) ) ) ) )
     => ( c_2Eiterate_2Ensum_2E2(A_27a,V0s_2E0,V2f_2E0) = c_2Eiterate_2Ensum_2E2(A_27b,V1t_2E0,V3g_2E0) ) ) ).

tff(thm_2Eiterate_2ENSUM__INJECTION,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1p_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
           => p(c_2Ebool_2EIN_2E2(A_27a,app_2E2(A_27a,A_27a,V1p_2E0,V3x_2E0),V2s_2E0)) )
        & ! [V4x_2E0: A_27a,V5y_2E0: A_27a] :
            ( ( p(c_2Ebool_2EIN_2E2(A_27a,V4x_2E0,V2s_2E0))
              & p(c_2Ebool_2EIN_2E2(A_27a,V5y_2E0,V2s_2E0))
              & ( app_2E2(A_27a,A_27a,V1p_2E0,V4x_2E0) = app_2E2(A_27a,A_27a,V1p_2E0,V5y_2E0) ) )
           => ( V4x_2E0 = V5y_2E0 ) ) )
     => ( c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Enum_2Enum,A_27a,V0f_2E0,V1p_2E0)) = c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V0f_2E0) ) ) ).

tff(thm_2Eiterate_2ENSUM__UNION__NONZERO,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V1s_2E0))
        & p(c_2Epred__set_2EFINITE_2E1(A_27a,V2t_2E0))
        & ! [V3x_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,c_2Epred__set_2EINTER_2E2(A_27a,V1s_2E0,V2t_2E0)))
           => ( app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3x_2E0) = c_2Enum_2E0_2E0 ) ) )
     => ( c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EUNION_2E2(A_27a,V1s_2E0,V2t_2E0),V0f_2E0) = c_2Earithmetic_2E_2B_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V1s_2E0,V0f_2E0),c_2Eiterate_2Ensum_2E2(A_27a,V2t_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ENSUM__BIGUNION__NONZERO,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Enum_2Enum))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V6t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Enum_2Enum),F0_2E0,V0f_2E0),V6t_2E0) = c_2Eiterate_2Ensum_2E2(A_27a,V6t_2E0,V0f_2E0) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1s_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] :
          ( ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1s_2E0))
            & ! [V2t_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
                ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2t_2E0,V1s_2E0))
               => p(c_2Epred__set_2EFINITE_2E1(A_27a,V2t_2E0)) )
            & ! [V3t1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4t2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V5x_2E0: A_27a] :
                ( ( p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3t1_2E0,V1s_2E0))
                  & p(c_2Ebool_2EIN_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4t2_2E0,V1s_2E0))
                  & ( V3t1_2E0 != V4t2_2E0 )
                  & p(c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V3t1_2E0))
                  & p(c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V4t2_2E0)) )
               => ( app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V5x_2E0) = c_2Enum_2E0_2E0 ) ) )
         => ( c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EBIGUNION_2E1(A_27a,V1s_2E0),V0f_2E0) = c_2Eiterate_2Ensum_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Enum_2Enum),F0_2E0,V0f_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__CASES,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)))),F4_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)))),F3_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))))] :
      ( ! [V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V3g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum))),F0_2E0,V1P_2E0),V2f_2E0),V3g_2E0),V4x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V4x_2E0),app_2E2(A_27a,tyop_2Enum_2Enum,V2f_2E0,V4x_2E0),app_2E2(A_27a,tyop_2Enum_2Enum,V3g_2E0,V4x_2E0)) )
     => ( ! [V6x_2E0: A_27a,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
            ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F4_2E0,V6x_2E0),V0s_2E0),V1P_2E0))
          <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V6x_2E0,V0s_2E0))
              & ~ p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V6x_2E0)) ) )
       => ( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V6x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F2_2E0,V0s_2E0),V1P_2E0),V6x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V6x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F4_2E0,V6x_2E0),V0s_2E0),V1P_2E0)) )
         => ( ! [V5x_2E0: A_27a,V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
                ( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F3_2E0,V5x_2E0),V0s_2E0),V1P_2E0))
              <=> ( p(c_2Ebool_2EIN_2E2(A_27a,V5x_2E0,V0s_2E0))
                  & p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V5x_2E0)) ) )
           => ( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V5x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V0s_2E0),V1P_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(A_27a,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),F3_2E0,V5x_2E0),V0s_2E0),V1P_2E0)) )
             => ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V3g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] :
                  ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
                 => ( c_2Eiterate_2Ensum_2E2(A_27a,V0s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum))),F0_2E0,V1P_2E0),V2f_2E0),V3g_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F1_2E0,V0s_2E0),V1P_2E0)),V2f_2E0),c_2Eiterate_2Ensum_2E2(A_27a,c_2Epred__set_2EGSPEC_2E1(A_27a,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F2_2E0,V0s_2E0),V1P_2E0)),V3g_2E0)) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__CLOSED,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Enum_2E0_2E0))
        & ! [V3x_2E0: tyop_2Enum_2Enum,V4y_2E0: tyop_2Enum_2Enum] :
            ( ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
              & p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,V4y_2E0)) )
           => p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Earithmetic_2E_2B_2E2(V3x_2E0,V4y_2E0))) )
        & ! [V5a_2E0: A_27a] :
            ( p(c_2Ebool_2EIN_2E2(A_27a,V5a_2E0,V2s_2E0))
           => p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,app_2E2(A_27a,tyop_2Enum_2Enum,V1f_2E0,V5a_2E0))) ) )
     => p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V0P_2E0,c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V1f_2E0))) ) ).

tff(thm_2Eiterate_2ENSUM__ADD__NUMSEG,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V4i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F0_2E0,V0f_2E0),V1g_2E0),V4i_2E0) = c_2Earithmetic_2E_2B_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V4i_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1g_2E0,V4i_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F0_2E0,V0f_2E0),V1g_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V0f_2E0),c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V1g_2E0)) ) ) ).

tff(thm_2Eiterate_2ENSUM__LE__NUMSEG,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
      ( ! [V4i_2E0: tyop_2Enum_2Enum] :
          ( ( p(c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V4i_2E0))
            & p(c_2Earithmetic_2E_3C_3D_2E2(V4i_2E0,V3n_2E0)) )
         => p(c_2Earithmetic_2E_3C_3D_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V4i_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1g_2E0,V4i_2E0))) )
     => p(c_2Earithmetic_2E_3C_3D_2E2(c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V0f_2E0),c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V1g_2E0))) ) ).

tff(thm_2Eiterate_2ENSUM__EQ__NUMSEG,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
      ( ! [V4i_2E0: tyop_2Enum_2Enum] :
          ( ( p(c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,V4i_2E0))
            & p(c_2Earithmetic_2E_3C_3D_2E2(V4i_2E0,V3n_2E0)) )
         => ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V4i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1g_2E0,V4i_2E0) ) )
     => ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V0f_2E0) = c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V1g_2E0) ) ) ).

tff(thm_2Eiterate_2ENSUM__CONST__NUMSEG,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))] :
      ( ! [V0c_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F0_2E0,V0c_2E0),V3n_2E0) = V0c_2E0 )
     => ! [V0c_2E0: tyop_2Enum_2Enum,V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F0_2E0,V0c_2E0)) = c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2B_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V1m_2E0),V0c_2E0) ) ) ).

tff(thm_2Eiterate_2ENSUM__EQ__0__NUMSEG,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
      ( ! [V3i_2E0: tyop_2Enum_2Enum] :
          ( ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V3i_2E0))
            & p(c_2Earithmetic_2E_3C_3D_2E2(V3i_2E0,V2n_2E0)) )
         => ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V3i_2E0) = c_2Enum_2E0_2E0 ) )
     => ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0) = c_2Enum_2E0_2E0 ) ) ).

tff(thm_2Eiterate_2ENSUM__EQ__0__IFF__NUMSEG,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
      ( ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0) = c_2Enum_2E0_2E0 )
    <=> ! [V3i_2E0: tyop_2Enum_2Enum] :
          ( ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V3i_2E0))
            & p(c_2Earithmetic_2E_3C_3D_2E2(V3i_2E0,V2n_2E0)) )
         => ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V3i_2E0) = c_2Enum_2E0_2E0 ) ) ) ).

tff(thm_2Eiterate_2ENSUM__TRIV__NUMSEG,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Eprim__rec_2E_3C_2E2(V2n_2E0,V1m_2E0))
     => ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0) = c_2Enum_2E0_2E0 ) ) ).

tff(thm_2Eiterate_2ENSUM__SING__NUMSEG,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1n_2E0,V1n_2E0),V0f_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V1n_2E0) ) ).

tff(thm_2Eiterate_2ENSUM__CLAUSES__NUMSEG,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
      ( ! [V1m_2E0: tyop_2Enum_2Enum] :
          ( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V1m_2E0))
        <=> ( V1m_2E0 = c_2Enum_2E0_2E0 ) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)] :
          ( ! [V1m_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,c_2Enum_2E0_2E0),V0f_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V1m_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,c_2Enum_2E0_2E0),c_2Enum_2E0_2E0) )
          & ! [V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)),V0f_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,c_2Earithmetic_2E_3C_3D_2E2(V2m_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)),c_2Earithmetic_2E_2B_2E2(c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V0f_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,c_2Enum_2ESUC_2E1(V3n_2E0))),c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),V0f_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__SWAP__NUMSEG,axiom,
    ! [F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))))] :
      ( ! [V4f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),V6j_2E0: tyop_2Enum_2Enum,V7i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F2_2E0,V4f_2E0),V6j_2E0),V7i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V4f_2E0,V7i_2E0),V6j_2E0) )
     => ( ! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum,V4f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),V6j_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))),F1_2E0,V0a_2E0),V1b_2E0),V4f_2E0),V6j_2E0) = c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V0a_2E0,V1b_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F2_2E0,V4f_2E0),V6j_2E0)) )
       => ( ! [V2c_2E0: tyop_2Enum_2Enum,V3d_2E0: tyop_2Enum_2Enum,V4f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),V5i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))),F0_2E0,V2c_2E0),V3d_2E0),V4f_2E0),V5i_2E0) = c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2c_2E0,V3d_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V4f_2E0,V5i_2E0)) )
         => ! [V0a_2E0: tyop_2Enum_2Enum,V1b_2E0: tyop_2Enum_2Enum,V2c_2E0: tyop_2Enum_2Enum,V3d_2E0: tyop_2Enum_2Enum,V4f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))] : ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V0a_2E0,V1b_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))),F0_2E0,V2c_2E0),V3d_2E0),V4f_2E0)) = c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2c_2E0,V3d_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))),F1_2E0,V0a_2E0),V1b_2E0),V4f_2E0)) ) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__ADD__SPLIT,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum,V3p_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,c_2Earithmetic_2E_2B_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))))
     => ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,c_2Earithmetic_2E_2B_2E2(V2n_2E0,V3p_2E0)),V0f_2E0) = c_2Earithmetic_2E_2B_2E2(c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0),c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2B_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Earithmetic_2E_2B_2E2(V2n_2E0,V3p_2E0)),V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ENSUM__OFFSET,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)))] :
      ( ! [V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V0p_2E0: tyop_2Enum_2Enum,V4i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F0_2E0,V1f_2E0),V0p_2E0),V4i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V1f_2E0,c_2Earithmetic_2E_2B_2E2(V4i_2E0,V0p_2E0)) )
     => ! [V0p_2E0: tyop_2Enum_2Enum,V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V2m_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2B_2E2(V2m_2E0,V0p_2E0),c_2Earithmetic_2E_2B_2E2(V3n_2E0,V0p_2E0)),V1f_2E0) = c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V2m_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F0_2E0,V1f_2E0),V0p_2E0)) ) ) ).

tff(thm_2Eiterate_2ENSUM__OFFSET__0,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1m_2E0: tyop_2Enum_2Enum,V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F0_2E0,V0f_2E0),V1m_2E0),V3i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,c_2Earithmetic_2E_2B_2E2(V3i_2E0,V1m_2E0)) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0))
         => ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0) = c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2E_2D_2E2(V2n_2E0,V1m_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F0_2E0,V0f_2E0),V1m_2E0)) ) ) ) ).

tff(thm_2Eiterate_2ENSUM__CLAUSES__LEFT,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
      ( p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0))
     => ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0) = c_2Earithmetic_2E_2B_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V1m_2E0),c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2B_2E2(V1m_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V2n_2E0),V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2ENSUM__CLAUSES__RIGHT,axiom,
    ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] :
      ( ( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,V2n_2E0))
        & p(c_2Earithmetic_2E_3C_3D_2E2(V1m_2E0,V2n_2E0)) )
     => ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),V0f_2E0) = c_2Earithmetic_2E_2B_2E2(c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,c_2Earithmetic_2E_2D_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0f_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V2n_2E0)) ) ) ).

tff(thm_2Eiterate_2ENSUM__PAIR,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F0_2E0,V0f_2E0),V3i_2E0) = c_2Earithmetic_2E_2B_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V3i_2E0)),app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V3i_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1m_2E0: tyop_2Enum_2Enum,V2n_2E0: tyop_2Enum_2Enum] : ( c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V1m_2E0),c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2A_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT2_2E1(c_2Earithmetic_2EZERO_2E0)),V2n_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),V0f_2E0) = c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1m_2E0,V2n_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),F0_2E0,V0f_2E0)) ) ) ).

tff(thm_2Eiterate_2EMOD__NSUM__MOD,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1n_2E0: tyop_2Enum_2Enum,V3i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V0f_2E0),V1n_2E0),V3i_2E0) = c_2Earithmetic_2EMOD_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V3i_2E0),V1n_2E0) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1n_2E0: tyop_2Enum_2Enum,V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
          ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V2s_2E0))
            & ( V1n_2E0 != c_2Enum_2E0_2E0 ) )
         => ( c_2Earithmetic_2EMOD_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,V0f_2E0),V1n_2E0) = c_2Earithmetic_2EMOD_2E2(c_2Eiterate_2Ensum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)),F0_2E0,V0f_2E0),V1n_2E0)),V1n_2E0) ) ) ) ).

tff(thm_2Eiterate_2EMOD__NSUM__MOD__NUMSEG,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)))] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V3n_2E0: tyop_2Enum_2Enum,V4i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F0_2E0,V0f_2E0),V3n_2E0),V4i_2E0) = c_2Earithmetic_2EMOD_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,V0f_2E0,V4i_2E0),V3n_2E0) )
     => ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),V1a_2E0: tyop_2Enum_2Enum,V2b_2E0: tyop_2Enum_2Enum,V3n_2E0: tyop_2Enum_2Enum] :
          ( ( V3n_2E0 != c_2Enum_2E0_2E0 )
         => ( c_2Earithmetic_2EMOD_2E2(c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1a_2E0,V2b_2E0),V0f_2E0),V3n_2E0) = c_2Earithmetic_2EMOD_2E2(c_2Eiterate_2Ensum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V1a_2E0,V2b_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F0_2E0,V0f_2E0),V3n_2E0)),V3n_2E0) ) ) ) ).

tff(thm_2Eiterate_2ETH,axiom,
    ! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F4_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal)),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
      ( ! [V5f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V10i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F1_2E0,V5f_2E0),V10i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V5f_2E0,V10i_2E0) )
     => ( ! [V11f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V16i_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),F4_2E0,V11f_2E0),V16i_2E0) = app_2E2(A_27b,tyop_2Erealax_2Ereal,V11f_2E0,V16i_2E0) )
       => ( ! [V13p_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V17y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),F3_2E0,V13p_2E0),V17y_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,V17y_2E0,app_2E2(A_27b,tyop_2Emin_2Ebool,V13p_2E0,V17y_2E0)) )
         => ( ! [V13p_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V15y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),F2_2E0,V13p_2E0),V15y_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,V15y_2E0,app_2E2(A_27b,tyop_2Emin_2Ebool,V13p_2E0,V15y_2E0)) )
           => ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V4i_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0),V4i_2E0) = app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V4i_2E0) )
             => ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V2s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
                    ( ! [V3x_2E0: A_27a] :
                        ( p(c_2Ebool_2EIN_2E2(A_27a,V3x_2E0,V2s_2E0))
                       => ( app_2E2(A_27a,tyop_2Erealax_2Ereal,V0f_2E0,V3x_2E0) = app_2E2(A_27a,tyop_2Erealax_2Ereal,V1g_2E0,V3x_2E0) ) )
                   => ( c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),F0_2E0,V0f_2E0)) = c_2Eiterate_2ESum_2E2(A_27a,V2s_2E0,V1g_2E0) ) )
                & ! [V5f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V6g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V7a_2E0: tyop_2Enum_2Enum,V8b_2E0: tyop_2Enum_2Enum] :
                    ( ! [V9i_2E0: tyop_2Enum_2Enum] :
                        ( ( p(c_2Earithmetic_2E_3C_3D_2E2(V7a_2E0,V9i_2E0))
                          & p(c_2Earithmetic_2E_3C_3D_2E2(V9i_2E0,V8b_2E0)) )
                       => ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V5f_2E0,V9i_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V6g_2E0,V9i_2E0) ) )
                   => ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V7a_2E0,V8b_2E0),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F1_2E0,V5f_2E0)) = c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(V7a_2E0,V8b_2E0),V6g_2E0) ) )
                & ! [V11f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V12g_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),V13p_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
                    ( ! [V14x_2E0: A_27b] :
                        ( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V13p_2E0,V14x_2E0))
                       => ( app_2E2(A_27b,tyop_2Erealax_2Ereal,V11f_2E0,V14x_2E0) = app_2E2(A_27b,tyop_2Erealax_2Ereal,V12g_2E0,V14x_2E0) ) )
                   => ( c_2Eiterate_2ESum_2E2(A_27b,c_2Epred__set_2EGSPEC_2E1(A_27b,A_27b,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),F2_2E0,V13p_2E0)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27b,tyop_2Erealax_2Ereal),F4_2E0,V11f_2E0)) = c_2Eiterate_2ESum_2E2(A_27b,c_2Epred__set_2EGSPEC_2E1(A_27b,A_27b,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),F3_2E0,V13p_2E0)),V12g_2E0) ) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2EREAL__SUB__POW,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))))] :
      ( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum,V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0,V0x_2E0),V1y_2E0),V2n_2E0),V3i_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Epow_2E2(V0x_2E0,V3i_2E0),c_2Ereal_2Epow_2E2(V1y_2E0,c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2D_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V3i_2E0))) )
     => ! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2n_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V2n_2E0))
         => ( c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Epow_2E2(V0x_2E0,V2n_2E0),c_2Ereal_2Epow_2E2(V1y_2E0,V2n_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1y_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2E_2D_2E2(V2n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F0_2E0,V0x_2E0),V1y_2E0),V2n_2E0))) ) ) ) ).

tff(thm_2Eiterate_2EREAL__SUB__POW__R1,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
      ( ! [V0x_2E0: tyop_2Erealax_2Ereal,V2i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V2i_2E0) = c_2Ereal_2Epow_2E2(V0x_2E0,V2i_2E0) )
     => ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V1n_2E0))
         => ( c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0),c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(V0x_2E0,c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2E_2D_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0))) ) ) ) ).

tff(thm_2Eiterate_2EREAL__SUB__POW__L1,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))] :
      ( ! [V0x_2E0: tyop_2Erealax_2Ereal,V2i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V2i_2E0) = c_2Ereal_2Epow_2E2(V0x_2E0,V2i_2E0) )
     => ! [V0x_2E0: tyop_2Erealax_2Ereal,V1n_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V1n_2E0))
         => ( c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),c_2Ereal_2Epow_2E2(V0x_2E0,V1n_2E0)) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V0x_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2E_2D_2E2(V1n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0))) ) ) ) ).

tff(thm_2Eiterate_2EFORALL__IN__GSPEC,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))] :
      ( ! [V14f_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),V16w_2E0: A_27e,V13P_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),V17x_2E0: A_27f,V18y_2E0: A_27g] : ( app_2E2(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F3_2E0,V14f_2E0),V16w_2E0),V13P_2E0),V17x_2E0),V18y_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27g,A_27b,app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,A_27b),app_2E2(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b)),V14f_2E0,V16w_2E0),V17x_2E0),V18y_2E0),app_2E2(A_27g,tyop_2Emin_2Ebool,app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),app_2E2(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool)),V13P_2E0,V16w_2E0),V17x_2E0),V18y_2E0)) )
     => ( ! [V14f_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),V13P_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),V16w_2E0: A_27e] : ( app_2E2(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F2_2E0,V14f_2E0),V13P_2E0),V16w_2E0) = c_2Epair_2EUNCURRY_2E1(A_27f,A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))))),F3_2E0,V14f_2E0),V16w_2E0),V13P_2E0)) )
       => ( ! [V7f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,A_27b)),V6P_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),V9x_2E0: A_27c,V10y_2E0: A_27d] : ( app_2E2(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F1_2E0,V7f_2E0),V6P_2E0),V9x_2E0),V10y_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27d,A_27b,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,A_27b),V7f_2E0,V9x_2E0),V10y_2E0),app_2E2(A_27d,tyop_2Emin_2Ebool,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),V6P_2E0,V9x_2E0),V10y_2E0)) )
         => ( ! [V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V2f_2E0),V1P_2E0),V4x_2E0) = c_2Epair_2E_2C_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,A_27b,V2f_2E0,V4x_2E0),app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V4x_2E0)) )
           => ! [V0Q_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
                ( ! [V1P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
                    ( ! [V3z_2E0: A_27b] :
                        ( p(c_2Ebool_2EIN_2E2(A_27b,V3z_2E0,c_2Epred__set_2EGSPEC_2E1(A_27b,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V2f_2E0),V1P_2E0))))
                       => p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,V3z_2E0)) )
                  <=> ! [V5x_2E0: A_27a] :
                        ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1P_2E0,V5x_2E0))
                       => p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,app_2E2(A_27a,A_27b,V2f_2E0,V5x_2E0))) ) )
                & ! [V6P_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),V7f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,A_27b))] :
                    ( ! [V8z_2E0: A_27b] :
                        ( p(c_2Ebool_2EIN_2E2(A_27b,V8z_2E0,c_2Epred__set_2EGSPEC_2E1(A_27b,tyop_2Epair_2Eprod(A_27c,A_27d),c_2Epair_2EUNCURRY_2E1(A_27c,A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F1_2E0,V7f_2E0),V6P_2E0)))))
                       => p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,V8z_2E0)) )
                  <=> ! [V11x_2E0: A_27c,V12y_2E0: A_27d] :
                        ( p(app_2E2(A_27d,tyop_2Emin_2Ebool,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Ebool),V6P_2E0,V11x_2E0),V12y_2E0))
                       => p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,app_2E2(A_27d,A_27b,app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,A_27b),V7f_2E0,V11x_2E0),V12y_2E0))) ) )
                & ! [V13P_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),V14f_2E0: tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b)))] :
                    ( ! [V15z_2E0: A_27b] :
                        ( p(c_2Ebool_2EIN_2E2(A_27b,V15z_2E0,c_2Epred__set_2EGSPEC_2E1(A_27b,tyop_2Epair_2Eprod(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g)),c_2Epair_2EUNCURRY_2E1(A_27e,tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27f,A_27g),tyop_2Epair_2Eprod(A_27b,tyop_2Emin_2Ebool)))),F2_2E0,V14f_2E0),V13P_2E0)))))
                       => p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,V15z_2E0)) )
                  <=> ! [V19w_2E0: A_27e,V20x_2E0: A_27f,V21y_2E0: A_27g] :
                        ( p(app_2E2(A_27g,tyop_2Emin_2Ebool,app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool),app_2E2(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,tyop_2Emin_2Ebool)),V13P_2E0,V19w_2E0),V20x_2E0),V21y_2E0))
                       => p(app_2E2(A_27b,tyop_2Emin_2Ebool,V0Q_2E0,app_2E2(A_27g,A_27b,app_2E2(A_27f,tyop_2Emin_2Efun(A_27g,A_27b),app_2E2(A_27e,tyop_2Emin_2Efun(A_27f,tyop_2Emin_2Efun(A_27g,A_27b)),V14f_2E0,V19w_2E0),V20x_2E0),V21y_2E0))) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2EREAL__SUB__POLYFUN,axiom,
    ! [F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
      ( ! [V0a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2y_2E0: tyop_2Erealax_2Ereal,V6j_2E0: tyop_2Enum_2Enum,V7i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F2_2E0,V0a_2E0),V2y_2E0),V6j_2E0),V7i_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0a_2E0,V7i_2E0),c_2Ereal_2Epow_2E2(V2y_2E0,c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2D_2E2(V7i_2E0,V6j_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))))) )
     => ( ! [V3n_2E0: tyop_2Enum_2Enum,V0a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V6j_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F1_2E0,V3n_2E0),V0a_2E0),V2y_2E0),V1x_2E0),V6j_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2E_2B_2E2(V6j_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0))),V3n_2E0),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F2_2E0,V0a_2E0),V2y_2E0),V6j_2E0)),c_2Ereal_2Epow_2E2(V1x_2E0,V6j_2E0)) )
       => ( ! [V0a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V4i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0a_2E0),V1x_2E0),V4i_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0a_2E0,V4i_2E0),c_2Ereal_2Epow_2E2(V1x_2E0,V4i_2E0)) )
         => ! [V0a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V3n_2E0: tyop_2Enum_2Enum] :
              ( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V3n_2E0))
             => ( c_2Ereal_2Ereal__sub_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V3n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0a_2E0),V1x_2E0)),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V3n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0a_2E0),V2y_2E0))) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(V1x_2E0,V2y_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2E_2D_2E2(V3n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F1_2E0,V3n_2E0),V0a_2E0),V2y_2E0),V1x_2E0))) ) ) ) ) ) ).

tff(thm_2Eiterate_2EREAL__SUB__POLYFUN__ALT,axiom,
    ! [F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
      ( ! [V0a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V6j_2E0: tyop_2Enum_2Enum,V2y_2E0: tyop_2Erealax_2Ereal,V7k_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F2_2E0,V0a_2E0),V6j_2E0),V2y_2E0),V7k_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0a_2E0,c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2E_2B_2E2(V6j_2E0,V7k_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),c_2Ereal_2Epow_2E2(V2y_2E0,V7k_2E0)) )
     => ( ! [V3n_2E0: tyop_2Enum_2Enum,V0a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2y_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal,V6j_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F1_2E0,V3n_2E0),V0a_2E0),V2y_2E0),V1x_2E0),V6j_2E0) = c_2Erealax_2Ereal__mul_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2E_2D_2E2(c_2Earithmetic_2E_2D_2E2(V3n_2E0,V6j_2E0),c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F2_2E0,V0a_2E0),V6j_2E0),V2y_2E0)),c_2Ereal_2Epow_2E2(V1x_2E0,V6j_2E0)) )
       => ( ! [V0a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V4i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0a_2E0),V1x_2E0),V4i_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V0a_2E0,V4i_2E0),c_2Ereal_2Epow_2E2(V1x_2E0,V4i_2E0)) )
         => ! [V0a_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal,V2y_2E0: tyop_2Erealax_2Ereal,V3n_2E0: tyop_2Enum_2Enum] :
              ( p(c_2Earithmetic_2E_3C_3D_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V3n_2E0))
             => ( c_2Ereal_2Ereal__sub_2E2(c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V3n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0a_2E0),V1x_2E0)),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V3n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V0a_2E0),V2y_2E0))) = c_2Erealax_2Ereal__mul_2E2(c_2Ereal_2Ereal__sub_2E2(V1x_2E0,V2y_2E0),c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,c_2Earithmetic_2E_2D_2E2(V3n_2E0,c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)))),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))),F1_2E0,V3n_2E0),V0a_2E0),V2y_2E0),V1x_2E0))) ) ) ) ) ) ).

tff(thm_2Eiterate_2EREAL__POLYFUN__ROOTBOUND,axiom,
    ! [F5_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F4_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)))),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
      ( ! [V1c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5x_2E0: tyop_2Erealax_2Ereal,V6i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F5_2E0,V1c_2E0),V5x_2E0),V6i_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1c_2E0,V6i_2E0),c_2Ereal_2Epow_2E2(V5x_2E0,V6i_2E0)) )
     => ( ! [V0n_2E0: tyop_2Enum_2Enum,V1c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5x_2E0: tyop_2Erealax_2Ereal] :
            ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F4_2E0,V0n_2E0),V1c_2E0),V5x_2E0))
          <=> ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V0n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F5_2E0,V1c_2E0),V5x_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
       => ( ! [V0n_2E0: tyop_2Enum_2Enum,V1c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V5x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0n_2E0),V1c_2E0),V5x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V5x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F4_2E0,V0n_2E0),V1c_2E0),V5x_2E0)) )
         => ( ! [V1c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal,V4i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F3_2E0,V1c_2E0),V3x_2E0),V4i_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1c_2E0,V4i_2E0),c_2Ereal_2Epow_2E2(V3x_2E0,V4i_2E0)) )
           => ( ! [V0n_2E0: tyop_2Enum_2Enum,V1c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] :
                  ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0n_2E0),V1c_2E0),V3x_2E0))
                <=> ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V0n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F3_2E0,V1c_2E0),V3x_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
             => ( ! [V0n_2E0: tyop_2Enum_2Enum,V1c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0n_2E0),V1c_2E0),V3x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V3x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F2_2E0,V0n_2E0),V1c_2E0),V3x_2E0)) )
               => ! [V0n_2E0: tyop_2Enum_2Enum,V1c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
                    ( ~ ! [V2i_2E0: tyop_2Enum_2Enum] :
                          ( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V2i_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V0n_2E0)))
                         => ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1c_2E0,V2i_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
                   => ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0n_2E0),V1c_2E0))))
                      & p(c_2Earithmetic_2E_3C_3D_2E2(c_2Epred__set_2ECARD_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F1_2E0,V0n_2E0),V1c_2E0))),V0n_2E0)) ) ) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2EREAL__POLYFUN__FINITE__ROOTS,axiom,
    ! [F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal))),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
      ( ! [V1c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal,V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F2_2E0,V1c_2E0),V2x_2E0),V3i_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1c_2E0,V3i_2E0),c_2Ereal_2Epow_2E2(V2x_2E0,V3i_2E0)) )
     => ( ! [V0n_2E0: tyop_2Enum_2Enum,V1c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] :
            ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0n_2E0),V1c_2E0),V2x_2E0))
          <=> ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V0n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F2_2E0,V1c_2E0),V2x_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) )
       => ( ! [V0n_2E0: tyop_2Enum_2Enum,V1c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0n_2E0),V1c_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0n_2E0),V1c_2E0),V2x_2E0)) )
         => ! [V0n_2E0: tyop_2Enum_2Enum,V1c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
              ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0n_2E0),V1c_2E0))))
            <=> ? [V4i_2E0: tyop_2Enum_2Enum] :
                  ( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4i_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V0n_2E0)))
                  & ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1c_2E0,V4i_2E0) != c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2EREAL__POLYFUN__EQ__0,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
      ( ! [V1c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal,V3i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V2x_2E0),V3i_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1c_2E0,V3i_2E0),c_2Ereal_2Epow_2E2(V2x_2E0,V3i_2E0)) )
     => ! [V0n_2E0: tyop_2Enum_2Enum,V1c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)] :
          ( ! [V2x_2E0: tyop_2Erealax_2Ereal] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V0n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V2x_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
        <=> ! [V4i_2E0: tyop_2Enum_2Enum] :
              ( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V4i_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V0n_2E0)))
             => ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1c_2E0,V4i_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ) ).

tff(thm_2Eiterate_2EREAL__POLYFUN__EQ__CONST,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)))] :
      ( ! [V1c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V3x_2E0: tyop_2Erealax_2Ereal,V4i_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V3x_2E0),V4i_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1c_2E0,V4i_2E0),c_2Ereal_2Epow_2E2(V3x_2E0,V4i_2E0)) )
     => ! [V0n_2E0: tyop_2Enum_2Enum,V1c_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),V2k_2E0: tyop_2Erealax_2Ereal] :
          ( ! [V3x_2E0: tyop_2Erealax_2Ereal] : ( c_2Eiterate_2ESum_2E2(tyop_2Enum_2Enum,c_2Eiterate_2E_2E_2E_2E2(c_2Enum_2E0_2E0,V0n_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V3x_2E0)) = V2k_2E0 )
        <=> ( ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1c_2E0,c_2Enum_2E0_2E0) = V2k_2E0 )
            & ! [V5i_2E0: tyop_2Enum_2Enum] :
                ( p(c_2Ebool_2EIN_2E2(tyop_2Enum_2Enum,V5i_2E0,c_2Eiterate_2E_2E_2E_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),V0n_2E0)))
               => ( app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,V1c_2E0,V5i_2E0) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ) ) ) ) ).

tff(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__CONST,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
      ( ! [V0c_2E0: tyop_2Erealax_2Ereal,V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0),V1x_2E0) = V0c_2E0 )
     => ! [V0c_2E0: tyop_2Erealax_2Ereal] : p(c_2Eiterate_2Epolynomial__function_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0c_2E0))) ) ).

tff(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__ID,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
      ( ! [V0x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V0x_2E0) = V0x_2E0 )
     => p(c_2Eiterate_2Epolynomial__function_2E1(F0_2E0)) ) ).

tff(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__ADD,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
      ( ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1q_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0p_2E0),V1q_2E0),V2x_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0p_2E0,V2x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1q_2E0,V2x_2E0)) )
     => ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1q_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
          ( ( p(c_2Eiterate_2Epolynomial__function_2E1(V0p_2E0))
            & p(c_2Eiterate_2Epolynomial__function_2E1(V1q_2E0)) )
         => p(c_2Eiterate_2Epolynomial__function_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0p_2E0),V1q_2E0))) ) ) ).

tff(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__LMUL,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
      ( ! [V1c_2E0: tyop_2Erealax_2Ereal,V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V0p_2E0),V2x_2E0) = c_2Erealax_2Ereal__mul_2E2(V1c_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0p_2E0,V2x_2E0)) )
     => ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1c_2E0: tyop_2Erealax_2Ereal] :
          ( p(c_2Eiterate_2Epolynomial__function_2E1(V0p_2E0))
         => p(c_2Eiterate_2Epolynomial__function_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1c_2E0),V0p_2E0))) ) ) ).

tff(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__RMUL,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
      ( ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1c_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0p_2E0),V1c_2E0),V2x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0p_2E0,V2x_2E0),V1c_2E0) )
     => ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1c_2E0: tyop_2Erealax_2Ereal] :
          ( p(c_2Eiterate_2Epolynomial__function_2E1(V0p_2E0))
         => p(c_2Eiterate_2Epolynomial__function_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0p_2E0),V1c_2E0))) ) ) ).

tff(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__NEG,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
      ( ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0p_2E0),V1x_2E0) = c_2Erealax_2Ereal__neg_2E1(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0p_2E0,V1x_2E0)) )
     => ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Eiterate_2Epolynomial__function_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0p_2E0)) = c_2Eiterate_2Epolynomial__function_2E1(V0p_2E0) ) ) ).

tff(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__SUB,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
      ( ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1q_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0p_2E0),V1q_2E0),V2x_2E0) = c_2Ereal_2Ereal__sub_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0p_2E0,V2x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1q_2E0,V2x_2E0)) )
     => ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1q_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
          ( ( p(c_2Eiterate_2Epolynomial__function_2E1(V0p_2E0))
            & p(c_2Eiterate_2Epolynomial__function_2E1(V1q_2E0)) )
         => p(c_2Eiterate_2Epolynomial__function_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0p_2E0),V1q_2E0))) ) ) ).

tff(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__MUL,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
      ( ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1q_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0p_2E0),V1q_2E0),V2x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0p_2E0,V2x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V1q_2E0,V2x_2E0)) )
     => ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1q_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
          ( ( p(c_2Eiterate_2Epolynomial__function_2E1(V0p_2E0))
            & p(c_2Eiterate_2Epolynomial__function_2E1(V1q_2E0)) )
         => p(c_2Eiterate_2Epolynomial__function_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0p_2E0),V1q_2E0))) ) ) ).

tff(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__SUM,axiom,
    ! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
      ( ! [V1p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V2i_2E0: A_27a,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1p_2E0),V2i_2E0),V3x_2E0) = app_2E2(A_27a,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1p_2E0,V3x_2E0),V2i_2E0) )
     => ( ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),V4x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0,V0s_2E0),V1p_2E0),V4x_2E0) = c_2Eiterate_2ESum_2E2(A_27a,V0s_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal),V1p_2E0,V4x_2E0)) )
       => ! [V0s_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal))] :
            ( ( p(c_2Epred__set_2EFINITE_2E1(A_27a,V0s_2E0))
              & ! [V2i_2E0: A_27a] :
                  ( p(c_2Ebool_2EIN_2E2(A_27a,V2i_2E0,V0s_2E0))
                 => p(c_2Eiterate_2Epolynomial__function_2E1(app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V1p_2E0),V2i_2E0))) ) )
           => p(c_2Eiterate_2Epolynomial__function_2E1(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(A_27a,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F1_2E0,V0s_2E0),V1p_2E0))) ) ) ) ).

tff(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__POW,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
      ( ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0p_2E0),V1n_2E0),V2x_2E0) = c_2Ereal_2Epow_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0p_2E0,V2x_2E0),V1n_2E0) )
     => ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1n_2E0: tyop_2Enum_2Enum] :
          ( p(c_2Eiterate_2Epolynomial__function_2E1(V0p_2E0))
         => p(c_2Eiterate_2Epolynomial__function_2E1(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F0_2E0,V0p_2E0),V1n_2E0))) ) ) ).

tff(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__INDUCT,axiom,
    ! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F1_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F3_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))),F2_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)))] :
      ( ! [V1x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0,V1x_2E0) = V1x_2E0 )
     => ( ! [V2c_2E0: tyop_2Erealax_2Ereal,V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F1_2E0,V2c_2E0),V3x_2E0) = V2c_2E0 )
       => ( ! [V7p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V8q_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V9x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F3_2E0,V7p_2E0),V8q_2E0),V9x_2E0) = c_2Erealax_2Ereal__mul_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V7p_2E0,V9x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V8q_2E0,V9x_2E0)) )
         => ( ! [V4p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5q_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V6x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F2_2E0,V4p_2E0),V5q_2E0),V6x_2E0) = c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V4p_2E0,V6x_2E0),app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V5q_2E0,V6x_2E0)) )
           => ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool)] :
                ( ( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V0P_2E0,F0_2E0))
                  & ! [V2c_2E0: tyop_2Erealax_2Ereal] : p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V0P_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F1_2E0,V2c_2E0)))
                  & ! [V4p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V5q_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
                      ( ( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V0P_2E0,V4p_2E0))
                        & p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V0P_2E0,V5q_2E0)) )
                     => p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V0P_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F2_2E0,V4p_2E0),V5q_2E0))) )
                  & ! [V7p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V8q_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
                      ( ( p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V0P_2E0,V7p_2E0))
                        & p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V0P_2E0,V8q_2E0)) )
                     => p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V0P_2E0,app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),F3_2E0,V7p_2E0),V8q_2E0))) ) )
               => ! [V10p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
                    ( p(c_2Eiterate_2Epolynomial__function_2E1(V10p_2E0))
                   => p(app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,V0P_2E0,V10p_2E0)) ) ) ) ) ) ) ).

tff(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__o,axiom,
    ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1q_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] :
      ( ( p(c_2Eiterate_2Epolynomial__function_2E1(V0p_2E0))
        & p(c_2Eiterate_2Epolynomial__function_2E1(V1q_2E0)) )
     => p(c_2Eiterate_2Epolynomial__function_2E1(c_2Ecombin_2Eo_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0p_2E0,V1q_2E0))) ) ).

tff(thm_2Eiterate_2EPOLYNOMIAL__FUNCTION__FINITE__ROOTS,axiom,
    ! [F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))))] :
      ( ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V2x_2E0: tyop_2Erealax_2Ereal,V1a_2E0: tyop_2Erealax_2Ereal] :
          ( p(app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0p_2E0),V2x_2E0),V1a_2E0))
        <=> ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0p_2E0,V2x_2E0) = V1a_2E0 ) )
     => ( ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal,V2x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0p_2E0),V1a_2E0),V2x_2E0) = c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,V2x_2E0,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),F1_2E0,V0p_2E0),V2x_2E0),V1a_2E0)) )
       => ! [V0p_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),V1a_2E0: tyop_2Erealax_2Ereal] :
            ( p(c_2Eiterate_2Epolynomial__function_2E1(V0p_2E0))
           => ( p(c_2Epred__set_2EFINITE_2E1(tyop_2Erealax_2Ereal,c_2Epred__set_2EGSPEC_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool))),F0_2E0,V0p_2E0),V1a_2E0))))
            <=> ~ ! [V3x_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0p_2E0,V3x_2E0) = V1a_2E0 ) ) ) ) ) ).

%------------------------------------------------------------------------------