ITP001 Axioms: ITP138^5.ax


%------------------------------------------------------------------------------
% File     : ITP138^5 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 set theory export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : real_sigma^2.ax [Gau20]
%          : HL4138^5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   36 (   0 unt;   1 typ;   0 def)
%            Number of atoms       : 1131 (  39 equ;   0 cnn)
%            Maximal formula atoms :   41 (  31 avg)
%            Number of connectives : 1782 (   6   ~;   0   |;  20   &;1607   @)
%                                         (   1 <=>; 148  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (  14 avg;1607 nst)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :    1 (   1   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   56 (  55 usr;  54 con; 0-2 aty)
%            Number of variables   :  163 (  26   ^ 136   !;   1   ?; 163   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Ereal__sigma_2EREAL__SUM__IMAGE,type,
    c_2Ereal__sigma_2EREAL__SUM__IMAGE: del > $i ).

thf(mem_c_2Ereal__sigma_2EREAL__SUM__IMAGE,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Erealax_2Ereal ) ) ) ).

thf(ax_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__DEF,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ V1s ) )
            = ( surj__ty_2Erealax_2Ereal
              @ ( ap
                @ ( ap
                  @ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ ty_2Erealax_2Ereal )
                    @ ( lam @ A_27a
                      @ ^ [V2e: $i] :
                          ( lam @ ty_2Erealax_2Ereal
                          @ ^ [V3acc: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V2e ) ) @ V3acc ) ) ) )
                  @ V1s )
                @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__THM,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
     => ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
          = ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
        & ! [V1e: $i] :
            ( ( mem @ V1e @ A_27a )
           => ! [V2s: $i] :
                ( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
               => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
                 => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1e ) @ V2s ) ) )
                    = ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V0f @ V1e ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V2s ) @ V1e ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__SING,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
     => ! [V1e: $i] :
          ( ( mem @ V1e @ A_27a )
         => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1e ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
            = ( surj__ty_2Erealax_2Ereal @ ( ap @ V0f @ V1e ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__POS,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
              & ! [V2x: $i] :
                  ( ( mem @ V2x @ A_27a )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
                   => ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ V2x ) ) ) ) ) )
           => ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ V1s ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__SPOS,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
          & ( V0s
           != ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
       => ! [V1f: $i] :
            ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
           => ( ! [V2x: $i] :
                  ( ( mem @ V2x @ A_27a )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
                   => ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) ) )
             => ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0s ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__NONZERO,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
       => ! [V1f: $i] :
            ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
           => ( ( ! [V2x: $i] :
                    ( ( mem @ V2x @ A_27a )
                   => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0P ) )
                     => ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) ) )
                & ? [V3x: $i] :
                    ( ( mem @ V3x @ A_27a )
                    & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0P ) )
                    & ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V3x ) )
                     != ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
             => ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) )
                 != ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
              <=> ( V0P
                 != ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IF__ELIM,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1P: $i] :
          ( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
         => ! [V2f: $i] :
              ( ( mem @ V2f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
             => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
                  & ! [V3x: $i] :
                      ( ( mem @ V3x @ A_27a )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
                       => ( p @ ( ap @ V1P @ V3x ) ) ) ) )
               => ( ( surj__ty_2Erealax_2Ereal
                    @ ( ap
                      @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
                        @ ( lam @ A_27a
                          @ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ V1P @ V4x ) ) @ ( ap @ V2f @ V4x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
                      @ V0s ) )
                  = ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f ) @ V0s ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__FINITE__SAME,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
       => ! [V1f: $i] :
            ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
           => ! [V2p: $i] :
                ( ( mem @ V2p @ A_27a )
               => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2p ) @ V0P ) )
                    & ! [V3q: $i] :
                        ( ( mem @ V3q @ A_27a )
                       => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3q ) @ V0P ) )
                         => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V2p ) )
                            = ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V3q ) ) ) ) ) )
                 => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) )
                    = ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0P ) ) ) @ ( ap @ V1f @ V2p ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__FINITE__CONST,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
       => ! [V1f: $i] :
            ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
           => ! [V2x: tp__ty_2Erealax_2Ereal] :
                ( ! [V3y: $i] :
                    ( ( mem @ V3y @ A_27a )
                   => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V3y ) )
                      = V2x ) )
               => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) )
                  = ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0P ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2x ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IN__IF,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
       => ! [V1f: $i] :
            ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
           => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) )
              = ( surj__ty_2Erealax_2Ereal
                @ ( ap
                  @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
                    @ ( lam @ A_27a
                      @ ^ [V2x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0P ) ) @ ( ap @ V1f @ V2x ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
                  @ V0P ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__CMUL,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
       => ! [V1f: $i] :
            ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
           => ! [V2c: tp__ty_2Erealax_2Ereal] :
                ( ( surj__ty_2Erealax_2Ereal
                  @ ( ap
                    @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
                      @ ( lam @ A_27a
                        @ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ V1f @ V3x ) ) ) )
                    @ V0P ) )
                = ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( inj__ty_2Erealax_2Ereal @ V2c ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__NEG,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
       => ! [V1f: $i] :
            ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
           => ( ( surj__ty_2Erealax_2Ereal
                @ ( ap
                  @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
                    @ ( lam @ A_27a
                      @ ^ [V2x: $i] : ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ V1f @ V2x ) ) ) )
                  @ V0P ) )
              = ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Ereal__neg @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IMAGE,axiom,
    ! [A_27a: del,A_27b: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
       => ! [V1f_27: $i] :
            ( ( mem @ V1f_27 @ ( arr @ A_27a @ A_27b ) )
           => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V1f_27 ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f_27 ) @ V0P ) ) )
             => ! [V2f: $i] :
                  ( ( mem @ V2f @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) )
                 => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f_27 ) @ V0P ) ) )
                    = ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Erealax_2Ereal @ A_27b ) @ V2f ) @ V1f_27 ) ) @ V0P ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__DISJOINT__UNION,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ! [V1P_27: $i] :
          ( ( mem @ V1P_27 @ ( arr @ A_27a @ bool ) )
         => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
              & ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1P_27 ) )
              & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0P ) @ V1P_27 ) ) )
           => ! [V2f: $i] :
                ( ( mem @ V2f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
               => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0P ) @ V1P_27 ) ) )
                  = ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f ) @ V0P ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f ) @ V1P_27 ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__EQ__CARD,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
       => ( ( surj__ty_2Erealax_2Ereal
            @ ( ap
              @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
                @ ( lam @ A_27a
                  @ ^ [V1x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0P ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
              @ V0P ) )
          = ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0P ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__INV__CARD__EQ__1,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ( ( ( V0P
           != ( c_2Epred__set_2EEMPTY @ A_27a ) )
          & ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) ) )
       => ( ( surj__ty_2Erealax_2Ereal
            @ ( ap
              @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
                @ ( lam @ A_27a
                  @ ^ [V1s: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1s ) @ V0P ) ) @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0P ) ) ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) )
              @ V0P ) )
          = ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__INTER__NONZERO,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
       => ! [V1f: $i] :
            ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
           => ( ( surj__ty_2Erealax_2Ereal
                @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f )
                  @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0P )
                    @ ( lam @ A_27a
                      @ ^ [V2p: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Erealax_2Ereal ) @ ( ap @ V1f @ V2p ) ) @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ) )
              = ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__INTER__ELIM,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
       => ! [V1f: $i] :
            ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
           => ! [V2P_27: $i] :
                ( ( mem @ V2P_27 @ ( arr @ A_27a @ bool ) )
               => ( ! [V3x: $i] :
                      ( ( mem @ V3x @ A_27a )
                     => ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2P_27 ) )
                       => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V3x ) )
                          = ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) )
                 => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0P ) @ V2P_27 ) ) )
                    = ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__COUNT,axiom,
    ! [V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
     => ! [V1n: tp__ty_2Enum_2Enum] :
          ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ty_2Enum_2Enum ) @ V0f ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
          = ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ V0f ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__MONO,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
       => ! [V1f: $i] :
            ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
           => ! [V2f_27: $i] :
                ( ( mem @ V2f_27 @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
               => ( ! [V3x: $i] :
                      ( ( mem @ V3x @ A_27a )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0P ) )
                       => ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2f_27 @ V3x ) ) ) ) )
                 => ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f_27 ) @ V0P ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__POS__MEM__LE,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
       => ! [V1f: $i] :
            ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
           => ( ! [V2x: $i] :
                  ( ( mem @ V2x @ A_27a )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0P ) )
                   => ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V1f @ V2x ) ) ) ) )
             => ! [V3x: $i] :
                  ( ( mem @ V3x @ A_27a )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0P ) )
                   => ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ V1f @ V3x ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__CONST__EQ__1__EQ__INV__CARD,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
       => ! [V1f: $i] :
            ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
           => ( ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) )
                  = ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
                & ! [V2x: $i] :
                    ( ( mem @ V2x @ A_27a )
                   => ! [V3y: $i] :
                        ( ( mem @ V3y @ A_27a )
                       => ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0P ) )
                            & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V0P ) ) )
                         => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V2x ) )
                            = ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V3y ) ) ) ) ) ) )
             => ! [V4x: $i] :
                  ( ( mem @ V4x @ A_27a )
                 => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V0P ) )
                   => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V4x ) )
                      = ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Erealax_2Einv @ ( ap @ c_2Ereal_2Ereal__of__num @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0P ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__ADD,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
       => ! [V1f: $i] :
            ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
           => ! [V2f_27: $i] :
                ( ( mem @ V2f_27 @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
               => ( ( surj__ty_2Erealax_2Ereal
                    @ ( ap
                      @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
                        @ ( lam @ A_27a
                          @ ^ [V3x: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2f_27 @ V3x ) ) ) )
                      @ V0s ) )
                  = ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f_27 ) @ V0s ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__REAL__SUM__IMAGE,axiom,
    ! [A_27a: del,A_27b: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1s_27: $i] :
          ( ( mem @ V1s_27 @ ( arr @ A_27b @ bool ) )
         => ! [V2f: $i] :
              ( ( mem @ V2f @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) ) )
             => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
                  & ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1s_27 ) ) )
               => ( ( surj__ty_2Erealax_2Ereal
                    @ ( ap
                      @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
                        @ ( lam @ A_27a
                          @ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b ) @ ( ap @ V2f @ V3x ) ) @ V1s_27 ) ) )
                      @ V0s ) )
                  = ( surj__ty_2Erealax_2Ereal
                    @ ( ap
                      @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
                        @ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
                          @ ^ [V4x: $i] : ( ap @ ( ap @ V2f @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V4x ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V4x ) ) ) )
                      @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1s_27 ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__0,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
       => ( ( surj__ty_2Erealax_2Ereal
            @ ( ap
              @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
                @ ( lam @ A_27a
                  @ ^ [V1x: $i] : ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
              @ V0s ) )
          = ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2ESEQ__REAL__SUM__IMAGE,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
       => ! [V1f: $i] :
            ( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) ) )
           => ! [V2f_27: $i] :
                ( ( mem @ V2f_27 @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
               => ( ! [V3x: $i] :
                      ( ( mem @ V3x @ A_27a )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
                       => ( p
                          @ ( ap
                            @ ( ap @ c_2Eseq_2E_2D_2D_3E
                              @ ( lam @ ty_2Enum_2Enum
                                @ ^ [V4n: $i] : ( ap @ ( ap @ V1f @ V4n ) @ V3x ) ) )
                            @ ( ap @ V2f_27 @ V3x ) ) ) ) )
                 => ( p
                    @ ( ap
                      @ ( ap @ c_2Eseq_2E_2D_2D_3E
                        @ ( lam @ ty_2Enum_2Enum
                          @ ^ [V5n: $i] : ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ ( ap @ V1f @ V5n ) ) @ V0s ) ) )
                      @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f_27 ) @ V0s ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2ENESTED__REAL__SUM__IMAGE__REVERSE,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ ty_2Erealax_2Ereal ) ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2s_27: $i] :
              ( ( mem @ V2s_27 @ ( arr @ A_27b @ bool ) )
             => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
                  & ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2s_27 ) ) )
               => ( ( surj__ty_2Erealax_2Ereal
                    @ ( ap
                      @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
                        @ ( lam @ A_27a
                          @ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ V2s_27 ) ) )
                      @ V1s ) )
                  = ( surj__ty_2Erealax_2Ereal
                    @ ( ap
                      @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27b )
                        @ ( lam @ A_27b
                          @ ^ [V4x: $i] :
                              ( ap
                              @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
                                @ ( lam @ A_27a
                                  @ ^ [V5y: $i] : ( ap @ ( ap @ V0f @ V5y ) @ V4x ) ) )
                              @ V1s ) ) )
                      @ V2s_27 ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__EQ__sum,axiom,
    ! [V0n: tp__ty_2Enum_2Enum,V1r: $i] :
      ( ( mem @ V1r @ ( arr @ ty_2Enum_2Enum @ ty_2Erealax_2Ereal ) )
     => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Esum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V1r ) )
        = ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ty_2Enum_2Enum ) @ V1r ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__POW,axiom,
    ! [A_27a: del,V0a: $i] :
      ( ( mem @ V0a @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
           => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Epow @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0a ) @ V1s ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
              = ( surj__ty_2Erealax_2Ereal
                @ ( ap
                  @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
                    @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ty_2Erealax_2Ereal )
                      @ ( lam @ A_27a
                        @ ^ [V2i: $i] :
                            ( lam @ A_27a
                            @ ^ [V3j: $i] : ( ap @ ( ap @ c_2Erealax_2Ereal__mul @ ( ap @ V0a @ V2i ) ) @ ( ap @ V0a @ V3j ) ) ) ) ) )
                  @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27a ) @ V1s ) @ V1s ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__EQ,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
         => ! [V2f_27: $i] :
              ( ( mem @ V2f_27 @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
             => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
                  & ! [V3x: $i] :
                      ( ( mem @ V3x @ A_27a )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
                       => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V1f @ V3x ) )
                          = ( surj__ty_2Erealax_2Ereal @ ( ap @ V2f_27 @ V3x ) ) ) ) ) )
               => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0s ) )
                  = ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f_27 ) @ V0s ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__IN__IF__ALT,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
         => ! [V2z: tp__ty_2Erealax_2Ereal] :
              ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
             => ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0s ) )
                = ( surj__ty_2Erealax_2Ereal
                  @ ( ap
                    @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
                      @ ( lam @ A_27a
                        @ ^ [V3x: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Erealax_2Ereal ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) ) @ ( ap @ V1f @ V3x ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) )
                    @ V0s ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__SUB,axiom,
    ! [A_27a: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
         => ! [V2f_27: $i] :
              ( ( mem @ V2f_27 @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
             => ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
               => ( ( surj__ty_2Erealax_2Ereal
                    @ ( ap
                      @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a )
                        @ ( lam @ A_27a
                          @ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ V1f @ V3x ) ) @ ( ap @ V2f_27 @ V3x ) ) ) )
                      @ V0s ) )
                  = ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V1f ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V2f_27 ) @ V0s ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__MONO__SET,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ty_2Erealax_2Ereal ) )
     => ! [V1s: $i] :
          ( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
         => ! [V2t: $i] :
              ( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
             => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
                  & ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2t ) )
                  & ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) )
                  & ! [V3x: $i] :
                      ( ( mem @ V3x @ A_27a )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2t ) )
                       => ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ V0f @ V3x ) ) ) ) ) )
               => ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ A_27a ) @ V0f ) @ V2t ) ) ) ) ) ) ) ).

thf(conj_thm_2Ereal__sigma_2EREAL__SUM__IMAGE__CROSS__SYM,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ty_2Erealax_2Ereal ) )
     => ! [V1s1: $i] :
          ( ( mem @ V1s1 @ ( arr @ A_27a @ bool ) )
         => ! [V2s2: $i] :
              ( ( mem @ V2s2 @ ( arr @ A_27b @ bool ) )
             => ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s1 ) )
                  & ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2s2 ) ) )
               => ( ( surj__ty_2Erealax_2Ereal
                    @ ( ap
                      @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
                        @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ty_2Erealax_2Ereal )
                          @ ( lam @ A_27a
                            @ ^ [V3x: $i] :
                                ( lam @ A_27b
                                @ ^ [V4y: $i] : ( ap @ V0f @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3x ) @ V4y ) ) ) ) ) )
                      @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V1s1 ) @ V2s2 ) ) )
                  = ( surj__ty_2Erealax_2Ereal
                    @ ( ap
                      @ ( ap @ ( c_2Ereal__sigma_2EREAL__SUM__IMAGE @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) )
                        @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27b @ A_27a @ ty_2Erealax_2Ereal )
                          @ ( lam @ A_27b
                            @ ^ [V5y: $i] :
                                ( lam @ A_27a
                                @ ^ [V6x: $i] : ( ap @ V0f @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V6x ) @ V5y ) ) ) ) ) )
                      @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27a ) @ V2s2 ) @ V1s1 ) ) ) ) ) ) ) ) ).

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