ITP001 Axioms: ITP008^5.ax


%------------------------------------------------------------------------------
% File     : ITP008^5 : 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    : combin^2.ax [Gau20]
%          : HL4008^5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   83 (   6 unt;  15 typ;   0 def)
%            Number of atoms       :  743 (  61 equ;   0 cnn)
%            Maximal formula atoms :   32 (   8 avg)
%            Number of connectives : 1653 (   5   ~;   1   |;   9   &;1490   @)
%                                         (  13 <=>; 135  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (  11 avg;1490 nst)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :   31 (  31   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   34 (  33 usr;  18 con; 0-3 aty)
%            Number of variables   :  302 (  25   ^ 277   !;   0   ?; 302   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Ecombin_2E_3A_3E,type,
    c_2Ecombin_2E_3A_3E: del > del > $i ).

thf(mem_c_2Ecombin_2E_3A_3E,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecombin_2E_3A_3E @ A_27a @ A_27b ) @ ( arr @ A_27b @ ( arr @ ( arr @ A_27b @ A_27a ) @ A_27a ) ) ) ).

thf(tp_c_2Ecombin_2EASSOC,type,
    c_2Ecombin_2EASSOC: del > $i ).

thf(mem_c_2Ecombin_2EASSOC,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Ecombin_2EASSOC @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ bool ) ) ).

thf(tp_c_2Ecombin_2EC,type,
    c_2Ecombin_2EC: del > del > del > $i ).

thf(mem_c_2Ecombin_2EC,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Ecombin_2EC @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) @ ( arr @ A_27b @ ( arr @ A_27a @ A_27c ) ) ) ) ).

thf(tp_c_2Ecombin_2ECOMM,type,
    c_2Ecombin_2ECOMM: del > del > $i ).

thf(mem_c_2Ecombin_2ECOMM,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecombin_2ECOMM @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) ) @ bool ) ) ).

thf(tp_c_2Ecombin_2EFAIL,type,
    c_2Ecombin_2EFAIL: del > del > $i ).

thf(mem_c_2Ecombin_2EFAIL,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecombin_2EFAIL @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ A_27b @ A_27a ) ) ) ).

thf(tp_c_2Ecombin_2EFCOMM,type,
    c_2Ecombin_2EFCOMM: del > del > del > $i ).

thf(mem_c_2Ecombin_2EFCOMM,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Ecombin_2EFCOMM @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ A_27a ) ) @ ( arr @ ( arr @ A_27c @ ( arr @ A_27a @ A_27a ) ) @ bool ) ) ) ).

thf(tp_c_2Ecombin_2EI,type,
    c_2Ecombin_2EI: del > $i ).

thf(mem_c_2Ecombin_2EI,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Ecombin_2EI @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) ).

thf(tp_c_2Ecombin_2EK,type,
    c_2Ecombin_2EK: del > del > $i ).

thf(mem_c_2Ecombin_2EK,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecombin_2EK @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ A_27b @ A_27a ) ) ) ).

thf(tp_c_2Ecombin_2ELEFT__ID,type,
    c_2Ecombin_2ELEFT__ID: del > del > $i ).

thf(mem_c_2Ecombin_2ELEFT__ID,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecombin_2ELEFT__ID @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) @ ( arr @ A_27a @ bool ) ) ) ).

thf(tp_c_2Ecombin_2EMONOID,type,
    c_2Ecombin_2EMONOID: del > $i ).

thf(mem_c_2Ecombin_2EMONOID,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Ecombin_2EMONOID @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ bool ) ) ) ).

thf(tp_c_2Ecombin_2ERIGHT__ID,type,
    c_2Ecombin_2ERIGHT__ID: del > del > $i ).

thf(mem_c_2Ecombin_2ERIGHT__ID,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecombin_2ERIGHT__ID @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ A_27a ) ) @ ( arr @ A_27b @ bool ) ) ) ).

thf(tp_c_2Ecombin_2ES,type,
    c_2Ecombin_2ES: del > del > del > $i ).

thf(mem_c_2Ecombin_2ES,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Ecombin_2ES @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27c ) ) ) ) ).

thf(tp_c_2Ecombin_2EUPDATE,type,
    c_2Ecombin_2EUPDATE: del > del > $i ).

thf(mem_c_2Ecombin_2EUPDATE,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) ) ) ) ).

thf(tp_c_2Ecombin_2EW,type,
    c_2Ecombin_2EW: del > del > $i ).

thf(mem_c_2Ecombin_2EW,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecombin_2EW @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) ) @ ( arr @ A_27a @ A_27b ) ) ) ).

thf(tp_c_2Ecombin_2Eo,type,
    c_2Ecombin_2Eo: del > del > del > $i ).

thf(mem_c_2Ecombin_2Eo,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27c @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27c ) @ ( arr @ A_27a @ A_27b ) ) ) ) ).

thf(ax_thm_2Ecombin_2EK__DEF,axiom,
    ! [A_27a: del,A_27b: del] :
      ( ( c_2Ecombin_2EK @ A_27a @ A_27b )
      = ( lam @ A_27a
        @ ^ [V0x: $i] :
            ( lam @ A_27b
            @ ^ [V1y: $i] : V0x ) ) ) ).

thf(ax_thm_2Ecombin_2ES__DEF,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del] :
      ( ( c_2Ecombin_2ES @ A_27a @ A_27b @ A_27c )
      = ( lam @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) )
        @ ^ [V0f: $i] :
            ( lam @ ( arr @ A_27a @ A_27b )
            @ ^ [V1g: $i] :
                ( lam @ A_27a
                @ ^ [V2x: $i] : ( ap @ ( ap @ V0f @ V2x ) @ ( ap @ V1g @ V2x ) ) ) ) ) ) ).

thf(ax_thm_2Ecombin_2EI__DEF,axiom,
    ! [A_27a: del] :
      ( ( c_2Ecombin_2EI @ A_27a )
      = ( ap @ ( ap @ ( c_2Ecombin_2ES @ A_27a @ ( arr @ A_27a @ A_27a ) @ A_27a ) @ ( c_2Ecombin_2EK @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) ) ) ).

thf(ax_thm_2Ecombin_2EC__DEF,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del] :
      ( ( c_2Ecombin_2EC @ A_27a @ A_27b @ A_27c )
      = ( lam @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) )
        @ ^ [V0f: $i] :
            ( lam @ A_27b
            @ ^ [V1x: $i] :
                ( lam @ A_27a
                @ ^ [V2y: $i] : ( ap @ ( ap @ V0f @ V2y ) @ V1x ) ) ) ) ) ).

thf(ax_thm_2Ecombin_2EW__DEF,axiom,
    ! [A_27a: del,A_27b: del] :
      ( ( c_2Ecombin_2EW @ A_27a @ A_27b )
      = ( lam @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) )
        @ ^ [V0f: $i] :
            ( lam @ A_27a
            @ ^ [V1x: $i] : ( ap @ ( ap @ V0f @ V1x ) @ V1x ) ) ) ) ).

thf(ax_thm_2Ecombin_2Eo__DEF,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27c @ A_27b ) )
     => ! [V1g: $i] :
          ( ( mem @ V1g @ ( arr @ A_27a @ A_27c ) )
         => ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g )
            = ( lam @ A_27a
              @ ^ [V2x: $i] : ( ap @ V0f @ ( ap @ V1g @ V2x ) ) ) ) ) ) ).

thf(ax_thm_2Ecombin_2EAPP__DEF,axiom,
    ! [A_27a: del,A_27b: del,V0x: $i] :
      ( ( mem @ V0x @ A_27b )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
         => ( ( ap @ ( ap @ ( c_2Ecombin_2E_3A_3E @ A_27a @ A_27b ) @ V0x ) @ V1f )
            = ( ap @ V1f @ V0x ) ) ) ) ).

thf(ax_thm_2Ecombin_2EUPDATE__def,axiom,
    ! [A_27a: del,A_27b: del,V0a: $i] :
      ( ( mem @ V0a @ A_27a )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ A_27b )
         => ( ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V0a ) @ V1b )
            = ( lam @ ( arr @ A_27a @ A_27b )
              @ ^ [V2f: $i] :
                  ( lam @ A_27a
                  @ ^ [V3c: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0a ) @ V3c ) ) @ V1b ) @ ( ap @ V2f @ V3c ) ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2Eo__THM,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
     => ! [V1g: $i] :
          ( ( mem @ V1g @ ( arr @ A_27c @ A_27a ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27c )
             => ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ A_27b @ A_27a ) @ V0f ) @ V1g ) @ V2x )
                = ( ap @ V0f @ ( ap @ V1g @ V2x ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2Eo__ASSOC,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
     => ! [V1g: $i] :
          ( ( mem @ V1g @ ( arr @ A_27c @ A_27a ) )
         => ! [V2h: $i] :
              ( ( mem @ V2h @ ( arr @ A_27d @ A_27c ) )
             => ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27d @ A_27b @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27d @ A_27a @ A_27c ) @ V1g ) @ V2h ) )
                = ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27d @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ A_27b @ A_27a ) @ V0f ) @ V1g ) ) @ V2h ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2Eo__ABS__L,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
     => ! [V1g: $i] :
          ( ( mem @ V1g @ ( arr @ A_27b @ A_27a ) )
         => ( ( ap
              @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ A_27c @ A_27a )
                @ ( lam @ A_27a
                  @ ^ [V2x: $i] : ( ap @ V0f @ V2x ) ) )
              @ V1g )
            = ( lam @ A_27b
              @ ^ [V3x: $i] : ( ap @ V0f @ ( ap @ V1g @ V3x ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2Eo__ABS__R,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27c @ A_27b ) )
     => ! [V1g: $i] :
          ( ( mem @ V1g @ ( arr @ A_27a @ A_27c ) )
         => ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27c ) @ V0f )
              @ ( lam @ A_27a
                @ ^ [V2x: $i] : ( ap @ V1g @ V2x ) ) )
            = ( lam @ A_27a
              @ ^ [V3x: $i] : ( ap @ V0f @ ( ap @ V1g @ V3x ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EK__THM,axiom,
    ! [A_27a: del,A_27b: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ A_27b )
         => ( ( ap @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27b ) @ V0x ) @ V1y )
            = V0x ) ) ) ).

thf(conj_thm_2Ecombin_2ES__THM,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
     => ! [V1g: $i] :
          ( ( mem @ V1g @ ( arr @ A_27a @ A_27b ) )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27a )
             => ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2ES @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g ) @ V2x )
                = ( ap @ ( ap @ V0f @ V2x ) @ ( ap @ V1g @ V2x ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2ES__ABS__L,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27c @ A_27b ) ) )
     => ! [V1g: $i] :
          ( ( mem @ V1g @ ( arr @ A_27a @ A_27c ) )
         => ( ( ap
              @ ( ap @ ( c_2Ecombin_2ES @ A_27a @ A_27c @ A_27b )
                @ ( lam @ A_27a
                  @ ^ [V2x: $i] : ( ap @ V0f @ V2x ) ) )
              @ V1g )
            = ( lam @ A_27a
              @ ^ [V3x: $i] : ( ap @ ( ap @ V0f @ V3x ) @ ( ap @ V1g @ V3x ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2ES__ABS__R,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27c @ A_27b ) ) )
     => ! [V1g: $i] :
          ( ( mem @ V1g @ ( arr @ A_27a @ A_27c ) )
         => ( ( ap @ ( ap @ ( c_2Ecombin_2ES @ A_27a @ A_27c @ A_27b ) @ V0f )
              @ ( lam @ A_27a
                @ ^ [V2x: $i] : ( ap @ V1g @ V2x ) ) )
            = ( lam @ A_27a
              @ ^ [V3x: $i] : ( ap @ ( ap @ V0f @ V3x ) @ ( ap @ V1g @ V3x ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EC__THM,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27b )
         => ! [V2y: $i] :
              ( ( mem @ V2y @ A_27a )
             => ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EC @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1x ) @ V2y )
                = ( ap @ ( ap @ V0f @ V2y ) @ V1x ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EC__ABS__L,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27c @ A_27b ) ) )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ A_27c )
         => ( ( ap
              @ ( ap @ ( c_2Ecombin_2EC @ A_27a @ A_27c @ A_27b )
                @ ( lam @ A_27a
                  @ ^ [V2x: $i] : ( ap @ V0f @ V2x ) ) )
              @ V1y )
            = ( lam @ A_27a
              @ ^ [V3x: $i] : ( ap @ ( ap @ V0f @ V3x ) @ V1y ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EW__THM,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ( ( ap @ ( ap @ ( c_2Ecombin_2EW @ A_27a @ A_27b ) @ V0f ) @ V1x )
            = ( ap @ ( ap @ V0f @ V1x ) @ V1x ) ) ) ) ).

thf(conj_thm_2Ecombin_2EI__THM,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ( ( ap @ ( c_2Ecombin_2EI @ A_27a ) @ V0x )
        = V0x ) ) ).

thf(conj_thm_2Ecombin_2EI__o__ID,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
     => ( ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27b ) @ ( c_2Ecombin_2EI @ A_27b ) ) @ V0f )
          = V0f )
        & ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ A_27a ) @ V0f ) @ ( c_2Ecombin_2EI @ A_27a ) )
          = V0f ) ) ) ).

thf(conj_thm_2Ecombin_2EK__o__THM,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del] :
      ( ! [V0f: $i] :
          ( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
         => ! [V1v: $i] :
              ( ( mem @ V1v @ A_27c )
             => ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ ( ap @ ( c_2Ecombin_2EK @ A_27c @ A_27b ) @ V1v ) ) @ V0f )
                = ( ap @ ( c_2Ecombin_2EK @ A_27c @ A_27a ) @ V1v ) ) ) )
      & ! [V2f: $i] :
          ( ( mem @ V2f @ ( arr @ A_27d @ A_27e ) )
         => ! [V3v: $i] :
              ( ( mem @ V3v @ A_27d )
             => ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27f @ A_27e @ A_27d ) @ V2f ) @ ( ap @ ( c_2Ecombin_2EK @ A_27d @ A_27f ) @ V3v ) )
                = ( ap @ ( c_2Ecombin_2EK @ A_27e @ A_27f ) @ ( ap @ V2f @ V3v ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EUPDATE__APPLY,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
      ( ! [V0a: $i] :
          ( ( mem @ V0a @ A_27a )
         => ! [V1x: $i] :
              ( ( mem @ V1x @ A_27b )
             => ! [V2f: $i] :
                  ( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
                 => ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V0a ) @ V1x ) @ V2f ) @ V0a )
                    = V1x ) ) ) )
      & ! [V3a: $i] :
          ( ( mem @ V3a @ A_27c )
         => ! [V4b: $i] :
              ( ( mem @ V4b @ A_27c )
             => ! [V5x: $i] :
                  ( ( mem @ V5x @ A_27d )
                 => ! [V6f: $i] :
                      ( ( mem @ V6f @ ( arr @ A_27c @ A_27d ) )
                     => ( ( V3a != V4b )
                       => ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27c @ A_27d ) @ V3a ) @ V5x ) @ V6f ) @ V4b )
                          = ( ap @ V6f @ V4b ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EAPPLY__UPDATE__THM,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ A_27a )
         => ! [V2b: $i] :
              ( ( mem @ V2b @ A_27b )
             => ! [V3c: $i] :
                  ( ( mem @ V3c @ A_27a )
                 => ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V2b ) @ V0f ) @ V3c )
                    = ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1a ) @ V3c ) ) @ V2b ) @ ( ap @ V0f @ V3c ) ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EUPDATE__COMMUTES,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ A_27a )
         => ! [V2b: $i] :
              ( ( mem @ V2b @ A_27a )
             => ! [V3c: $i] :
                  ( ( mem @ V3c @ A_27b )
                 => ! [V4d: $i] :
                      ( ( mem @ V4d @ A_27b )
                     => ( ( V1a != V2b )
                       => ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V3c ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V2b ) @ V4d ) @ V0f ) )
                          = ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V2b ) @ V4d ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V3c ) @ V0f ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EUPDATE__EQ,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ A_27a )
         => ! [V2b: $i] :
              ( ( mem @ V2b @ A_27b )
             => ! [V3c: $i] :
                  ( ( mem @ V3c @ A_27b )
                 => ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V3c ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V2b ) @ V0f ) )
                    = ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V3c ) @ V0f ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EUPDATE__APPLY__ID,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ A_27a )
         => ! [V2b: $i] :
              ( ( mem @ V2b @ A_27b )
             => ( ( ( ap @ V0f @ V1a )
                  = V2b )
              <=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V2b ) @ V0f )
                  = V0f ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EUPDATE__APPLY__IMP__ID,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
     => ! [V1b: $i] :
          ( ( mem @ V1b @ A_27b )
         => ! [V2a: $i] :
              ( ( mem @ V2a @ A_27a )
             => ( ( ( ap @ V0f @ V2a )
                  = V1b )
               => ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V2a ) @ V1b ) @ V0f )
                  = V0f ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EAPPLY__UPDATE__ID,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ A_27a )
         => ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ ( ap @ V0f @ V1a ) ) @ V0f )
            = V0f ) ) ) ).

thf(conj_thm_2Ecombin_2EUPD11__SAME__BASE,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ A_27a )
         => ! [V2b: $i] :
              ( ( mem @ V2b @ A_27a )
             => ! [V3c: $i] :
                  ( ( mem @ V3c @ A_27b )
                 => ! [V4d: $i] :
                      ( ( mem @ V4d @ A_27b )
                     => ( ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V3c ) @ V0f )
                          = ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V2b ) @ V4d ) @ V0f ) )
                      <=> ( ( ( V1a = V2b )
                            & ( V3c = V4d ) )
                          | ( ( V1a != V2b )
                            & ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V3c ) @ V0f )
                              = V0f )
                            & ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V2b ) @ V4d ) @ V0f )
                              = V0f ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2ESAME__KEY__UPDATE__DIFFER,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27c @ A_27d ) )
     => ! [V1f1: $i] :
          ( ( mem @ V1f1 @ A_27a )
         => ! [V2f2: $i] :
              ( ( mem @ V2f2 @ A_27b )
             => ! [V3a: $i] :
                  ( ( mem @ V3a @ A_27c )
                 => ! [V4b: $i] :
                      ( ( mem @ V4b @ A_27d )
                     => ! [V5c: $i] :
                          ( ( mem @ V5c @ A_27d )
                         => ( ( V4b != V5c )
                           => ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27c @ A_27d ) @ V3a ) @ V4b ) @ V0f )
                             != ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27c @ A_27d ) @ V3a ) @ V5c ) @ V0f ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EUPD11__SAME__KEY__AND__BASE,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ A_27a )
         => ! [V2b: $i] :
              ( ( mem @ V2b @ A_27b )
             => ! [V3c: $i] :
                  ( ( mem @ V3c @ A_27b )
                 => ( ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V2b ) @ V0f )
                      = ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V1a ) @ V3c ) @ V0f ) )
                  <=> ( V2b = V3c ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EUPD__SAME__KEY__UNWIND,axiom,
    ! [A_27a: del,A_27b: del,V0f1: $i] :
      ( ( mem @ V0f1 @ ( arr @ A_27a @ A_27b ) )
     => ! [V1f2: $i] :
          ( ( mem @ V1f2 @ ( arr @ A_27a @ A_27b ) )
         => ! [V2a: $i] :
              ( ( mem @ V2a @ A_27a )
             => ! [V3b: $i] :
                  ( ( mem @ V3b @ A_27b )
                 => ! [V4c: $i] :
                      ( ( mem @ V4c @ A_27b )
                     => ( ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V2a ) @ V3b ) @ V0f1 )
                          = ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V2a ) @ V4c ) @ V1f2 ) )
                       => ( ( V3b = V4c )
                          & ! [V5v: $i] :
                              ( ( mem @ V5v @ A_27b )
                             => ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V2a ) @ V5v ) @ V0f1 )
                                = ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V2a ) @ V5v ) @ V1f2 ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EGEN__LET__RAND,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27b @ A_27a ) )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27c @ A_27b ) )
         => ! [V2v: $i] :
              ( ( mem @ V2v @ A_27c )
             => ( ( ap @ V0P @ ( ap @ ( ap @ ( c_2Ebool_2ELET @ A_27c @ A_27b ) @ V1f ) @ V2v ) )
                = ( ap @ ( ap @ ( c_2Ebool_2ELET @ A_27c @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ A_27a @ A_27b ) @ V0P ) @ V1f ) ) @ V2v ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EGEN__LET__RATOR,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
     => ! [V1v: $i] :
          ( ( mem @ V1v @ A_27b )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27c )
             => ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ELET @ A_27b @ ( arr @ A_27c @ A_27a ) ) @ V0f ) @ V1v ) @ V2x )
                = ( ap @ ( ap @ ( c_2Ebool_2ELET @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2EC @ A_27b @ A_27c @ A_27a ) @ V0f ) @ V2x ) ) @ V1v ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2ELET__FORALL__ELIM,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ bool ) )
     => ! [V1v: $i] :
          ( ( mem @ V1v @ A_27a )
         => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ELET @ A_27a @ bool ) @ V0f ) @ V1v ) )
          <=> ( p @ ( ap @ ( c_2Ebool_2E_21 @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2ES @ A_27a @ bool @ bool ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ( arr @ bool @ bool ) @ bool ) @ c_2Emin_2E_3D_3D_3E ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ bool @ bool ) @ c_2Emarker_2EAbbrev ) @ ( ap @ ( ap @ ( c_2Ecombin_2EC @ A_27a @ A_27a @ bool ) @ ( c_2Emin_2E_3D @ A_27a ) ) @ V1v ) ) ) ) @ V0f ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EGEN__literal__case__RAND,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27b @ A_27a ) )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27c @ A_27b ) )
         => ! [V2v: $i] :
              ( ( mem @ V2v @ A_27c )
             => ( ( ap @ V0P @ ( ap @ ( ap @ ( c_2Ebool_2Eliteral__case @ A_27c @ A_27b ) @ V1f ) @ V2v ) )
                = ( ap @ ( ap @ ( c_2Ebool_2Eliteral__case @ A_27c @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ A_27a @ A_27b ) @ V0P ) @ V1f ) ) @ V2v ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EGEN__literal__case__RATOR,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
     => ! [V1v: $i] :
          ( ( mem @ V1v @ A_27b )
         => ! [V2x: $i] :
              ( ( mem @ V2x @ A_27c )
             => ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2Eliteral__case @ A_27b @ ( arr @ A_27c @ A_27a ) ) @ V0f ) @ V1v ) @ V2x )
                = ( ap @ ( ap @ ( c_2Ebool_2Eliteral__case @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2EC @ A_27b @ A_27c @ A_27a ) @ V0f ) @ V2x ) ) @ V1v ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2Eliteral__case__FORALL__ELIM,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ bool ) )
     => ! [V1v: $i] :
          ( ( mem @ V1v @ A_27a )
         => ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2Eliteral__case @ A_27a @ bool ) @ V0f ) @ V1v ) )
          <=> ( p @ ( ap @ ( c_2Ebool_2E_21 @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2ES @ A_27a @ bool @ bool ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ( arr @ bool @ bool ) @ bool ) @ c_2Emin_2E_3D_3D_3E ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ bool @ bool ) @ c_2Emarker_2EAbbrev ) @ ( ap @ ( ap @ ( c_2Ecombin_2EC @ A_27a @ A_27a @ bool ) @ ( c_2Emin_2E_3D @ A_27a ) ) @ V1v ) ) ) ) @ V0f ) ) ) ) ) ) ).

thf(ax_thm_2Ecombin_2EASSOC__DEF,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
     => ( ( p @ ( ap @ ( c_2Ecombin_2EASSOC @ A_27a ) @ V0f ) )
      <=> ! [V1x: $i] :
            ( ( mem @ V1x @ A_27a )
           => ! [V2y: $i] :
                ( ( mem @ V2y @ A_27a )
               => ! [V3z: $i] :
                    ( ( mem @ V3z @ A_27a )
                   => ( ( ap @ ( ap @ V0f @ V1x ) @ ( ap @ ( ap @ V0f @ V2y ) @ V3z ) )
                      = ( ap @ ( ap @ V0f @ ( ap @ ( ap @ V0f @ V1x ) @ V2y ) ) @ V3z ) ) ) ) ) ) ) ).

thf(ax_thm_2Ecombin_2ECOMM__DEF,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) ) )
     => ( ( p @ ( ap @ ( c_2Ecombin_2ECOMM @ A_27a @ A_27b ) @ V0f ) )
      <=> ! [V1x: $i] :
            ( ( mem @ V1x @ A_27a )
           => ! [V2y: $i] :
                ( ( mem @ V2y @ A_27a )
               => ( ( ap @ ( ap @ V0f @ V1x ) @ V2y )
                  = ( ap @ ( ap @ V0f @ V2y ) @ V1x ) ) ) ) ) ) ).

thf(ax_thm_2Ecombin_2EFCOMM__DEF,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27a ) ) )
     => ! [V1g: $i] :
          ( ( mem @ V1g @ ( arr @ A_27c @ ( arr @ A_27a @ A_27a ) ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Ecombin_2EFCOMM @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g ) )
          <=> ! [V2x: $i] :
                ( ( mem @ V2x @ A_27c )
               => ! [V3y: $i] :
                    ( ( mem @ V3y @ A_27a )
                   => ! [V4z: $i] :
                        ( ( mem @ V4z @ A_27b )
                       => ( ( ap @ ( ap @ V1g @ V2x ) @ ( ap @ ( ap @ V0f @ V3y ) @ V4z ) )
                          = ( ap @ ( ap @ V0f @ ( ap @ ( ap @ V1g @ V2x ) @ V3y ) ) @ V4z ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Ecombin_2ERIGHT__ID__DEF,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27a ) ) )
     => ! [V1e: $i] :
          ( ( mem @ V1e @ A_27b )
         => ( ( p @ ( ap @ ( ap @ ( c_2Ecombin_2ERIGHT__ID @ A_27a @ A_27b ) @ V0f ) @ V1e ) )
          <=> ! [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
               => ( ( ap @ ( ap @ V0f @ V2x ) @ V1e )
                  = V2x ) ) ) ) ) ).

thf(ax_thm_2Ecombin_2ELEFT__ID__DEF,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
     => ! [V1e: $i] :
          ( ( mem @ V1e @ A_27a )
         => ( ( p @ ( ap @ ( ap @ ( c_2Ecombin_2ELEFT__ID @ A_27a @ A_27b ) @ V0f ) @ V1e ) )
          <=> ! [V2x: $i] :
                ( ( mem @ V2x @ A_27b )
               => ( ( ap @ ( ap @ V0f @ V1e ) @ V2x )
                  = V2x ) ) ) ) ) ).

thf(ax_thm_2Ecombin_2EMONOID__DEF,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
     => ! [V1e: $i] :
          ( ( mem @ V1e @ A_27a )
         => ( ( p @ ( ap @ ( ap @ ( c_2Ecombin_2EMONOID @ A_27a ) @ V0f ) @ V1e ) )
          <=> ( ( p @ ( ap @ ( c_2Ecombin_2EASSOC @ A_27a ) @ V0f ) )
              & ( p @ ( ap @ ( ap @ ( c_2Ecombin_2ERIGHT__ID @ A_27a @ A_27a ) @ V0f ) @ V1e ) )
              & ( p @ ( ap @ ( ap @ ( c_2Ecombin_2ELEFT__ID @ A_27a @ A_27a ) @ V0f ) @ V1e ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EASSOC__CONJ,axiom,
    p @ ( ap @ ( c_2Ecombin_2EASSOC @ bool ) @ c_2Ebool_2E_2F_5C ) ).

thf(conj_thm_2Ecombin_2EASSOC__SYM,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
     => ( ( p @ ( ap @ ( c_2Ecombin_2EASSOC @ A_27a ) @ V0f ) )
      <=> ! [V1x: $i] :
            ( ( mem @ V1x @ A_27a )
           => ! [V2y: $i] :
                ( ( mem @ V2y @ A_27a )
               => ! [V3z: $i] :
                    ( ( mem @ V3z @ A_27a )
                   => ( ( ap @ ( ap @ V0f @ ( ap @ ( ap @ V0f @ V1x ) @ V2y ) ) @ V3z )
                      = ( ap @ ( ap @ V0f @ V1x ) @ ( ap @ ( ap @ V0f @ V2y ) @ V3z ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EASSOC__DISJ,axiom,
    p @ ( ap @ ( c_2Ecombin_2EASSOC @ bool ) @ c_2Ebool_2E_5C_2F ) ).

thf(conj_thm_2Ecombin_2EFCOMM__ASSOC,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
     => ( ( p @ ( ap @ ( ap @ ( c_2Ecombin_2EFCOMM @ A_27a @ A_27a @ A_27a ) @ V0f ) @ V0f ) )
      <=> ( p @ ( ap @ ( c_2Ecombin_2EASSOC @ A_27a ) @ V0f ) ) ) ) ).

thf(conj_thm_2Ecombin_2EMONOID__CONJ__T,axiom,
    p @ ( ap @ ( ap @ ( c_2Ecombin_2EMONOID @ bool ) @ c_2Ebool_2E_2F_5C ) @ c_2Ebool_2ET ) ).

thf(conj_thm_2Ecombin_2EMONOID__DISJ__F,axiom,
    p @ ( ap @ ( ap @ ( c_2Ecombin_2EMONOID @ bool ) @ c_2Ebool_2E_5C_2F ) @ c_2Ebool_2EF ) ).

thf(ax_thm_2Ecombin_2EFAIL__DEF,axiom,
    ! [A_27a: del,A_27b: del] :
      ( ( c_2Ecombin_2EFAIL @ A_27a @ A_27b )
      = ( lam @ A_27a
        @ ^ [V0x: $i] :
            ( lam @ A_27b
            @ ^ [V1y: $i] : V0x ) ) ) ).

thf(conj_thm_2Ecombin_2EFAIL__THM,axiom,
    ! [A_27a: del,A_27b: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1y: $i] :
          ( ( mem @ V1y @ A_27b )
         => ( ( ap @ ( ap @ ( c_2Ecombin_2EFAIL @ A_27a @ A_27b ) @ V0x ) @ V1y )
            = V0x ) ) ) ).

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